An application of generalized Morrey spaces to unique continuation property of the quasilinear elliptic equations

1.   Introduction

Labeling graphs have attracted the attention of numerous researchers in different disciplines. The importance of this research line is in fact due to the following:

(1) Finding different coding techniques for securing the communication networks and database managements.

(2) Providing a high-level secrecy to military services which is the most important factor for coding.

(3) Providing confidentiality, and integrity of messages transferred between group members which is a critical networking issue. For more application, see ^[1,2].

Coding through special kinds of graphs with different kinds of labeling is structured by many papers. Coding with Fibonacci web graph using super mean labeling was introduced by Uma Maheswari et al. ^[3]. Prasad et al. developed a technique of coding secret messages using sun flower graphs $SF_{n}$. Furthermore, every labeling graph can be converted to a code by using GMJ coding methods (see, ^[4] and the references therein)

A graph $\; G \;$ is a pair $\; (V, E), \;$ where $\; V(G)$ and $E(G)\;$ denote the vertex set and edge set of a graph $\; G$. The position of the vertices and the length of the edges do not concern us, what is important is the size of a graph (number of vertices) and the pairs of vertices which are connected by an edge. If $\; e = \{u, v \}\;$ is an edge of a graph $\; G, \;$ then $\; u\;$ and $\; v\;$ are adjacent while $\; u\;$ and $\; e\;$ are incident. Let $\; q = \; | E(G) |\;$ be the cardinality of $\; E(G)\;$ and $p\; = | V(G) |$ be that of $\; V(G)$. For every vertex $u \in V(G)$, the open neighborhood set $N(u)$ is the set of all vertices adjacent to $u$ in $\; G$. Graph operations^[5] allow us to generate many new graphs from old ones. A fan graph $F_{n}$ is defined as the join $\; P_{n}+ K_{1}\;$ where $\; P_{n}\;$ is the path graph on $n$ vertices and $K_{1}$ is a complete graph on one vertex. The crown graph $Cr_{n}$(Sunlet graph) is the graph on $2n$ vertices obtained by attaching $n\;$ pendant edges to a cycle graph $C_{n}$, i.e., the coronas $C_{n} \odot K_{1}$.

Definition 1.1. (i) The splitting graph $\; S^{\prime}(G)\;$ of a connected graph $\; G$ is the graph obtained by adding new vertex $\; u_{i}$ corresponding to each vertex $v_{i}\;$ of $\; V(G)$ such that $N(v_{i}) = N(u_{i})$.

(ii) The shadow graph $\; D_{2}(G)\;$ of a connected graph $\; G\;$ is constructed by taking two copies of $\; G$, say $\; G_{1}\;$ and $\; G_{2}\;$. Join each vertex $\; v_{i}\;$ in $\; G_{1}\;$ to the neighbors of the corresponding vertex $\; u_{i} \;$ in $\; G_{2}$. The shadow graph $\; D_{2}(G)\;$ can be obtained from the splitting graph $\; S^{\prime}(G)\;$ by adding edge between any two new vertices $u_{i}$ and $u_{j}$ if the corresponding original vertices $v_{i}$ and $v_{j}$ are adjacent. $V[D_{2}(G)] = V[S^{\prime}(G)]$, $\; E[D_{2}(G)] = E[S^{\prime}(G)]\cup \{u_{i} u_{i+1}, \; i = 1, 2, \; \cdots, n\; \}$.

(iii) The middle graph $M(G)$ of a connected graph $\; G\;$ is the graph whose vertex set is $\; V (G) \cup E(G) \;$ and in which two vertices in $M(G)$ are adjacent whenever either they are adjacent edges of $\; G\;$ or one is a vertex of $\; G\;$ and the other is an edge incident with it.

(iv) The total graph $T(G)$ of a connected graph $G$ is a graph such that the vertex set of $T(G)$ corresponds to the vertices and edges of $\; G\;$ and two vertices are adjacent in $T(G)$ if their corresponding elements are either adjacent or incident in $G$. The total graph $T(G)$ can be obtained from the middle graph $M(G)$ by adding edge between any two original vertices $v_{i}$ and $v_{j}$ if the corresponding new vertices $u_{i}$ and $u_{j}$ are adjacent. $V[T(G)] = V[M(G)]$, $\; \; \; E[T(G)] = E[M(G)]\cup E(G)$.

There are many different kinds of graph labeling^{[6,7,8,9,10,11,12,13]}, all that kinds of labeling problem will have following three common characteristics. A set of numbers from which vertex or edge labels are chosen, A rule that assigns a value to each edge or vertex, A condition that these values must satisfy. For a comprehensive survey on graph labeling refers to a dynamic survey of graph labeling ^[14].

Zeen El Deen ^[15] introduced the edge $\; \delta -$ graceful labeling of graphs by using the numbers $\; \{ \delta, \; 2 \delta, \; 3 \delta,$ $\; \cdots\; , \; q\delta\; \}$ to label the edges of a graph, for any positive integer $\; \delta$. He showed edge $\; \delta-$ graceful labeling for some graphs related to cycles.

Definition 1.2. An edge $\; \delta-$ graceful labeling of a graph $\; G = (V(G), \; E(G))\;$ with $\; p = |V(G)|$ vertices and $q = |E(G)|$ edges is a bijective mapping $\; f\;$ of the edge set $\; E(G)\;$ into the set $\; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \}$ such that the induced mapping $\; f^{\ast} :V(G)\; \rightarrow \{0, \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , k \delta-\delta\}$, given by: $f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k)$, where $k = max (p, q)$, is an injective function. The graph that admits an edge $\; \delta-$ graceful labeling is called an edge $\; \delta-$ graceful graph, if $\; \delta = \; 2\;$ we have the edge even labeling also if $\; \delta = \; 3\;$ we have the edge triple labeling and so on.

Example 1.1. In Figure 1 we present an edge $\; \delta-$ graceful labeling of a tree graph on $p = 11\;$ vertices and $\; q = 10\;$ edges $\; f:E(G)\; \rightarrow \{\delta, \; 2 \delta, 3\; \delta, \; \cdots, \; 10\; \delta\}\;$ and $\; f^{\ast} :V(G)\; \rightarrow \{0, \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , 10\delta\}$, given by: $f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (11\delta)$.

Example 1.2. Duplication of an edge $\; e = (xy)\;$ by a new vertex $\; u\;$ in a graph $\; G$ produces a new graph $H\;$ such that $\; N(u) = \{x, y\}$. Let $\{ v_{1}, v_{2}, \; \cdots, \; v_{n} \}\;$ be the vertex set in the path $P_{n}$ and $\; G\;$ be the graph obtained by duplication of each edge $\; v_{i}v_{i+1} \;$ of path $P_{n}$ by vertex $\; u_{i}, \; (1 \leq i < n)\;$. Then $\; V (G) = V (P_{n}) \cup \{ u_{1}, \; u_{2}, \; \cdots, \; u_{n-1}\}\;$ and the edge set are $\{ v_{i}v_{i+1}, \; v_{i}u_{i}, \; u_{i}v_{i+1}, \; i = 1, 2, \; \cdots\; , n-1\; \}\;$, so the graph $\; G\;$ has $p = \; 2n-1$ vertices and $\; q = 3n-3$ edges, $k = max (p, q)\; = \; 3n-3$. An edge $5-$ graceful labeling of a graph obtained duplication of each edge of $\; P_{11}\;$ by a vertex is shown in Figure 2.

2.   Edge $\; \delta-$ graceful labeling of some splitting graphs

2.1.   Edge $\; \delta-$ graceful labeling of the splitting graph $\; S^{\prime}(C_{n})\;$

Theorem 2.1. For any positive integer $\delta\;$, the splitting graph $\; S^{\prime}(C_{n})\;$ of the cycle $C_{n}, \; n > 3\;$ is an edge $\; \delta-$ graceful graph.

Proof. Let $\{ v_{1}, v_{2}, \; \cdots, \; v_{n} \}\;$ be the vertices of $C_{n}$ where these vertices are in their natural order module $n$. To form the splitting graph $\; S^{\prime}(C_{n})\;$ we add new vertices $\{ u_{1}, u_{2}, \; \cdots, \; u_{n} \; \}$ corresponding to vertices of $C_{n}$. The edges set of $S^{\prime}(C_{n})$ are $\{ v_{i}v_{i+1}, \; v_{i}u_{i+1}, \; u_{i}v_{i+1}, \; i = 1, 2, \; \cdots\; , n\; \}$, the graph $\; S^{\prime}(C_{n})\;$ has $p = \; 2n$ vertices and $\; q = 3n$ edges, $k = max (p, q)\; = \; 3n$.

Case (1): When $\; n \equiv\; 2\; mod \; 4, \; \; n > 3$. We define the labeling function $f: E(S^{\prime}(C_{n})\; ) \longrightarrow \{\delta, \; 2\; \delta, \cdots, \; (3n)\delta \}$ as follows:

In view of the above labeling pattern then the induced vertex labels are:

$\; \; \; \; f^{\ast}(v_{1}) = \; (n+1)\; \delta, \; \; \; \; \; \; \; \; \; f^{\ast}(v_{2}) = \; (2n+4)\; \delta, \; \; \; \; f^{\ast}(u_{1}) = \; 0, \; \; \; \; \; \; \; f^{\ast}(u_{n}) = (n-1)\; \delta, \;$ and

$\; \; f^{\ast}(v_{i}) = \; (2n+4i-4)\; \delta \; mod\; (3n \delta), \; \; \; \; \; \; \; \; \; \; \; for\; \; \; 3\leq \; i\; \leq \; n$,

$\; \; f^{\ast}(u_{i}) = \; (n+2i-1)\; \delta\; mod\; (3n \delta), \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n-1$.

Hence the vertex labels are all distinct and a multiple of $\; \delta$.

Case (2): When $\; n \equiv\; 0\; mod \; 4, \; \; n = 4$, the graph $S^{\prime}(C_{4})$ is an edge $\delta-$ graceful graph for any positive integer $\delta$ define the labeling function $f: E(S^{\prime}(C_{4})\; ) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; (12)\delta \}$ as shown in Figure 3.

$\; \bullet\;$ When $\; n \equiv\; 0\; mod \; 4, \; \; n > 4$, we define the labeling function $f$ as follows:

In view of the above labeling pattern then the induced vertex labels are:

$\; f^{\ast}(v_{1}) = \; (n+1)\; \delta, \; \; \; \; f^{\ast}(v_{n-2}) = \; (3n-7)\; \delta, \; \; \; \; \; f^{\ast}(v_{n-1}) = \; (3n-3)\; \delta, \; \; \; \; \; \; f^{\ast}(v_{n}) = (2n-3)\; \delta, \;$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{i}) = \; (2n+4i-1)\; \delta\; mod\; (3n \delta), \; \; \; \; \; \; \; \; \; \; \; \; for \; \; \; \; \; 2 \leq \; i\; \leq \; n-3$,

$f^{\ast}(u_{1}) = \; (n+2)\; \delta, \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{i}) = \; (n+2i)\; \delta \; mod\; (3n \delta), \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; \; 2 \leq \; i\; \leq \; n$.

Hence the vertex labels are all distinct and a multiple of $\; \delta$.

Case (3): When $n$ is odd, we define the labeling function $f$ as follows:

In view of the above labeling pattern then the induced vertex labels are:

Hence the vertex labels are all distinct and a multiple of $\; \delta$. Therefore $S^{\prime}(C_{n})\;$ admits an edge $\; \delta-$ graceful labeling.

Illustration: The splitting graphs $\; S^{\prime}(C_{8})\;$ with an edge $4-$ graceful labeling, $\; S^{\prime}(C_{9})\;$ with an edge $5-$ graceful labeling and splitting graph $\; S^{\prime}(C_{10})\;$ with an edge $3-$ graceful labeling are shown in Figure 4.

2.2.   Edge $\; \delta-$ graceful labeling of the splitting graph $\; S^{\prime}(F_{n})\;$

Theorem 2.2. For any positive integer $\delta$, the splitting graph $\; S^{\prime}(F_{n})\;$ of the fan graph $F_{n}$ is an edge $\; \delta-$ graceful graph.

Proof. Let $\{\; v_{0}, \; v_{1}, \; v_{2}, \; \cdots, \; v_{n} \}\;$ be the vertices of the fan $F_{n}$, to form the splitting graph $\; S^{\prime}(F_{n})\;$ we add new vertices $\{u_{0}, \; u_{1}, \; u_{2}, \; \cdots, \; u_{n} \; \}$ corresponding to the vertices of the fan $F_{n}$. The edges set of $\; S^{\prime}(F_{n})\;$ are $\{ v_{i}u_{i+1}, \; v_{i}v_{i+1}, \; u_{i}v_{i+1}, \; \; 1 \leq i \leq n-1\; \}\; \cup \{u_{i}v_{0}\; , \; v_{i}u_{0}, \; v_{i}v_{0}\; , \; \; 1 \leq i \leq n\; \}$, the graph $\; S^{\prime}(F_{n})\;$ has $p = \; 2n+2$ vertices and $\; q = 6n-3$ edges, $k = max (p, q)\; = \; 6n-3$, see Figure 5.

Case (1): When $n$ is even, we define the labeling function $f: E(S^{\prime}(F_{n})\; ) \longrightarrow \{\delta, \; 2\delta, \; \cdots, \; (6n-3)\delta\; \}$ as follows:

In view of the above labeling pattern then the induced vertex labels are:

$\; \; \; \; \; \; \; f^{\ast}(v_{0}) = \; 0\; , \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{1}) = \; (5n-2)\; \delta, \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{n}) = \; \; \delta, \;$

$\; \; \; \; f^{\ast}(u_{1}) = \; (n)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{n}) = (2n-2)\; \delta, \; \; \;$

$\; \; \; \; f^{\ast}(v_{i}) = \; \; (4n-3+2i)\; \delta \; mod\; [(6n-3)\delta], \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n -1, \; \; \;$

$\; \; \; f^{\ast}(u_{i}) = \; \; \; (3n-2+i)\; \delta\; mod\; [(6n-3)\delta], \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n -1, \; \; \;$

Finally, $\; f^{\ast}(u_{0}) = \; [\sum\limits_{i = 1}^{n-1} f(u_{0} v_{i})+f(u_{0} v_{n})]\; mod(6n-3)\; \delta = [\sum\limits_{i = 1}^{n-1}(3n-2+i)\; \delta]\; mod(6n-3)\; \delta$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; = [(\dfrac{7 n^{2}}{2} -\dfrac{11 n}{2}+2)\; \delta]\; mod(6n-3)\; \delta$.

If $\; n\equiv2 \; mod\; 12 \; \; \Longrightarrow\; \; n = 12k+2$

$\; \; f^{\ast}(u_{0}) = [(504 k^{2}+102k+5)\; \delta]\; mod(72k+9)\; \delta = \; [(39 k+5)\; \delta]\; mod(72k+9)\; \delta$

$\; \; \; \; \; \; \; \; \; \; = \; [\; (\dfrac{13n-6}{4})\; \delta]\; mod(6n-3)\; \delta$.

Similarly, If $\; n\equiv0 \; mod\; 12 \; \; \Longrightarrow\; \; n = 12k\; \;$ then $\; \; f^{\ast}(u_{0}) = \; [(\dfrac{9n-4}{4})\; \delta]\; mod(6n-3)\; \delta$.

If $\; n\equiv4 \; mod\; 12 \; \; \Longrightarrow\; \; n = 12k+4\; \;$ then $\; \; f^{\ast}(u_{0}) = \; [\; (\dfrac{17n-8}{4}) \; \delta]\; mod(6n-3)\; \delta$.

If $\; \; \; n\equiv6 \; mod\; 12 \; \; \Longrightarrow\; \; n = 12k+6\; \;$, then $\; \; f^{\ast}(u_{0}) = \; [\; (\dfrac{21n-10}{4}) \; \delta]\; mod(6n-3)\; \delta$.

If $\; n\equiv8 \; mod\; 12 \; \; \Longrightarrow\; \; n = 12k+8\; \;$ then $\; \; f^{\ast}(u_{0}) = \; [\; (\dfrac{n}{4}) \; \delta]\; mod(6n-3)\; \delta$.

If $\; n\equiv10 \; mod\; 12 \; \; \Longrightarrow\; \; n = 12k+10\; \;$ then $\; \; f^{\ast}(u_{0}) = \; [\; (\dfrac{5n-2}{4}) \; \delta]\; mod(6n-3)\; \delta$.

Case (2): When $n$ is odd, $\; \; n > 3$, we define the labeling function $f$ as follows:

In view of the above labeling pattern then the induced vertex labels are:

$\; \; \; \; f^{\ast}(v_{0}) = \; 0, \; \; \; \; \; \; \; \; \; f^{\ast}(v_{1}) = \; (\dfrac{7n+1}{2})\; \delta, \; \; \; \; f^{\ast}(v_{2}) = \; (4)\; \delta, \; \; \; \; \; \; \; f^{\ast}(v_{n}) = \; (4n-1)\; \delta$,

$\; \; \; \; f^{\ast}(u_{1}) = \; (3n-2)\; \delta, \; \; \; \; \; \; \; f^{\ast}(u_{n-1}) = (6n-5)\; \delta, \; \; \; \; \; \; \; f^{\ast}(u_{n}) = (\dfrac{7n-5}{2})\; \delta$,

$f^{\ast}(v_{i}) = \; (\dfrac{37}{2}n-\dfrac{17}{2}+i)\; \delta \; mod\; [(6n-3)\delta] = (\dfrac{n+1}{2}+i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 3 \leq \; i\; \leq \; n -1, \; \; \;$

$f^{\ast}(u_{i}) = \; (\dfrac{15n-9}{2}+i)\; \delta\; mod\; [(6n-3)\delta] = (\dfrac{3n-3}{2}+i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n -2, \; \; \;$

Finally, $\; \; \; \; f^{\ast}(u_{0}) = \sum_{i = 1}^{n} f(u_{0} v_{i}) = [\sum_{i = 2}^{n-1} f(u_{0} v_{i})+f(u_{0} v_{1})+f(u_{0} v_{n})]\; mod(6n-3)\; \delta$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; = [\sum_{i = 2}^{n-1}(\dfrac{5}{2}n-\dfrac{3}{2}+i)\; \delta+(5n-2)\; \delta]\; mod(6n-3)\; \delta = \; [(3 n^{2}-2n)\; \delta]\; mod(6n-3)\; \delta$

$\; \; \because\; n\equiv1 \; mod\; 2 \; \; \Longrightarrow\; \; n = 2k+1$

$\; \; f^{\ast}(u_{0}) = [(12 k^{2}+8k+1)\; \delta]\; mod(12k+3)\; \delta\; = \; [(5 k+1)\; \delta]\; mod(12k+3)\; \delta\; = \; (\dfrac{5n-3}{2})\; \delta$.

Hence the vertex labels are all distinct and a multiple of $\; \delta$. Therefore $S^{\prime}(F_{n})\;$ admits an edge $\; \delta-$ graceful labeling.

Illustration: The splitting graphs $\; S^{\prime}(F_{8})$ with an edge $5-$ graceful labeling and$\; S^{\prime}(F_{9})$ with an edge $4-$ graceful labeling are presented in Figure 6.

2.3.   Edge $\delta-$graceful labeling of the splitting graph $\; S^{\prime}(Cr_{n})\;$ of the crown graph

Theorem 2.3. For any positive integer $\delta$, the splitting graph $\; S^{\prime}(Cr_{n})\;$ of the crown graph is an edge $\; \delta-$ graceful graph.

Proof. Let $\{ v_{1}, v_{2}, \; \cdots, \; v_{n} \; \}$ and $\{ u_{1}, u_{2}, \; \cdots, \; u_{n} \; \}$ be the vertices of the crown $Cr_{n}$, to form the splitting graph $\; S^{\prime}(Cr_{n})\;$ we add newly vertices $\{ v^{\prime}_{1}, v^{\prime}_{2}, \; \cdots, \; v^{\prime}_{n} \}\;$ and $\{ u^{\prime}_{1}, u^{\prime}_{2}, \; \cdots, \; u^{\prime}_{n} \; \}$ corresponding to the vertices of the crown $Cr_{n}$. The edges set of the splitting graph $\; S^{\prime}(Cr_{n})\;$ are $\{ v_{i}\; u^{\prime}_{i}, \; v^{\prime}_{i}\; v_{i+1 },$ $\; v_{i}\; v^{\prime}_{i+1}, \; v_{i}\; v_{i+1}, \; v^{\prime}_{i}\; u_{i}, \; u_{i}\; v_{i }, \; \; 1 \leq i \leq n\; \}, \;$ see Figure 7. The graph $\; S^{\prime}(Cr_{n})\;$ has $p = \; 4n$ vertices and $\; q = 6n$ edges, $k = max (p, q)\; = \; 6n$.

Case (1): When $n$ is odd. We define the labeling function $f: E(S^{\prime}(Cr_{n})\; ) \longrightarrow \{\delta, \; 2\delta, \; \cdots, \; (6n)\delta \}$ as follows:

In view of the above labeling pattern then the induced vertex labels are:

$\; \; f^{\ast}(u^{\prime}_{1}) = \; n\; \delta, \; \; \; \; \; f^{\ast}(u^{\prime}_{n}) = \; \delta, \; \; \; \; f^{\ast}(v^{\prime}_{1}) = \; 0, \; \; \; \; f^{\ast}(v_{1}) = \; (6n-1)\; \delta, \; \; \; \; \; \; \; f^{\ast}(v_{n}) = \; (2n-1)\; \delta$,

$\; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{i}) = \; (5n-2i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 1 \leq \; i\; \leq \; n, \; \;$

$\; \; \; \; \; \; \; \; \; \; f^{\ast}(u^{\prime}_{i}) = \; (i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n-1, \; \; \; \;$

$\; \; \; \; \; \; \; \; \; \; f^{\ast}(v^{\prime}_{i}) = \; (3n-i+1)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\leq \; i\; \leq \; n, \; \;$

$\; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{i}) = \; (4n+2i-2)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n-1.\; \;$

Case (2): When $\; n\;$ is even. The labeling function $f: E(S^{\prime}(Cr_{n})\; ) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; (6n)\delta \}$ defined as follows:

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f(u_{i}\; v_{i }) = \delta\; (i), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 1 \leq \; i\; \leq \; n, \; \;$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f(v^{\prime}_{i}\; v_{i+1 }) = \delta\; (3n+i), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 1 \leq \; i\; \leq \; n, \; \;$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f(v_{i}\; v^{\prime}_{i+1}) = \delta\; (5n+1-i), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 1 \leq \; i\; \leq \; n, \; \;$

$f(v_{1}\; u^{\prime}_{1}) = \; \delta\; (6n), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f(v_{i}\; u^{\prime}_{i}) = \delta \; (2n+1-i), \; \; \; \; \; for\; \; \; 2 \leq \; i\; \leq \; n, \; \;$

$f(v^{\prime}_{1}\; u_{1}) = \; \delta\; (6n-2), \; \; \; \; \; \; \; \; \; \; \; \; \; f(v^{\prime}_{i}\; u_{i}) = \; \delta \; (2n+i), \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n, \; \;$

In view of the above labeling pattern then the induced vertex labels are:

$\; \; \; \; f^{\ast}(v_{1}) = (4n+2)\; \delta, \; \; \; \; \; f^{\ast}(v_{\frac{n}{2}+1}) = \delta, \; \; \; \; \; f^{\ast}(v_{\frac{n}{2}+2}) = (5n+4)\delta, \; \; \; \; \; f^{\ast}(v_{n}) = (6n-3)\delta, \;$

$\; \; \; \; \; f^{\ast}(u_{1}) = (6n-1)\; \delta, \; \; \; \; \; \; f^{\ast}(u^{\prime}_{1}) = 0, \; \; \; \; \; f^{\ast}(v^{\prime}_{1}) = (n)\; \delta, \;$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{i}) = (2n+2i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n, \; \;$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u^{\prime}_{i}) = (2n+1-i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n, \; \;$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v^{\prime}_{i}) = (4n+i+2)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; n, \; \;$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{i}) = (2n+4i-3)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq \; i\; \leq \; \dfrac{n}{2}, \; \;$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{i}) = (4i-5)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; \; \frac{n}{2}+3 \leq \; i\; \leq \; n-1.\; \;$

Hence the vertex labels are all distinct. Therefore $S^{\prime}(Cr_{n})\;$ admits an edge $\; \delta-$ graceful labeling.

Illustration: The splitting graphs $\; S^{\prime}(Cr_{8})\;$ of the crown graph with an edge $5-$ graceful labeling and $\; S^{\prime}(Cr_{9})\;$ with an edge $3-$ graceful labeling are shown in Figure 8.

3.   Edge $\delta-$ graceful labeling of the shadow graph of some graphs

3.1.   Edge $\delta-$ graceful labeling of the shadow graph $D_{2}(P_{n})$

Theorem 3.1. For any positive integer $\delta$, the shadow graph $\; D_{2}(P_{n}), \; \; n > 2\; \;$ of the path $P_{n}$ is an edge $\; \delta-$ graceful graph.

Proof. Let $\{ v_{1}, v_{2}, \; \cdots, \; v_{n} \}\;$ be the vertices in first copy of $P_{n}$ and $\{ u_{1}, u_{2}, \; \cdots, \; u_{n} \}\;$ be that in second copy of $P_{n}$. The edges set in the shadow graph $\; D_{2}(P_{n})$ are $\{ v_{i}v_{i+1}, \; v_{i}u_{i+1}, \; u_{i}v_{i+1}, \; u_{i}u_{i+1}, \; \; 1 \leq i \leq n-1\; \}$. The four vertices $v_{1}, \; v_{n}, \; u_{1}\;$ and $u_{n}$ are of degree $2$ and the remaining vertices are of degree $4$, so the graph $\; D_{2}(P_{n})\;$ has $p = 2n$ vertices and $q = 4n-4$ edges, $k = max (p, q)\; = \; 4n-4$.

$\; \bullet\;$ If $\; n = 2\;$ the graph $D_{2}(P_{2})$ is not an edge $\delta-$ graceful graph since it isomorphic to $\; C_{4}$ ^[2].

$\; \bullet\;$ If $\; n = 3\;$ the graph $D_{2}(P_{3})$ is an edge $\delta-$ graceful graph for any positive integer $\delta$ define the labeling function $f: E(D_{2}(P_{3})) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; 8\delta \}$ as shown in Figure 9.

$\; \bullet\;$ If $\; n \geq 4$. We define the labeling function $f: E(D_{2}(P_{n})) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; (4n -4)\delta \}$ as follows:

In view of the above labeling pattern then the induced vertex labels are:

$f^{\ast}(v_{1}) = \; 0, \; \; \; \; \; f^{\ast}(v_{n}) = \; \delta, \; \; \; \; \; f^{\ast}(u_{1}) = \; (2n-2)\; \delta, \; \; \; \; \; f^{\ast}(u_{n}) = (4n-5)\; \delta,$

$f^{\ast}(u_{2}) = \; 2\; \delta, \; \; \; \; \; \; \; f^{\ast}(u_{3}) = \; (2n+1)\; \delta, \; \; \; \; \; \; \; f^{\ast}(u_{4}) = \; (2n+2)\; \delta,$

$f^{\ast}(v_{i}) = [f(v_{i}v_{i+1})+ f(v_{i-1}v_{i})+f(v_{i}u_{i-1})+f(v_{i}u_{i+1})] \; mod\; [(4n-4)\delta]$

$\; \; \; \; \; \; \; \; = \; (2n+1-2i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; for\; \; \; 2 \leq i \leq n-1.$

Similarly, $\; f^{\ast}(u_{i}) = \; (2n-5+2i)\; \delta, \; for\; 5 \leq i \leq n-1.$

Hence the labels of the vertices are all distinct numbers and a multiple of $\; \delta$. Thus $\; D_{2}(P_{n})\;$ is an edge $\; \delta-$ graceful graph.

Illustration: The shadow graph $\; D_{2}(P_{13})$ with edge $\; 3-$ graceful labeling is presented in Figure 10.

3.2.   Edge $\delta-$ graceful labeling of the shadow graph $\; D_{2}(C_{n})\;$ of the cycle $C_{n}$

Theorem 3.2. For any positive integer $\delta\;$, the shadow graph $\; D_{2}(C_{n}), \; \; n\geq 3\;$ of the cycle $C_{n}$ is an edge $\; \delta-$ graceful graph.

Proof. Let $\{ v_{1}, v_{2}, \; \cdots, \; v_{n} \}\;$ be the vertices in first copy of $C_{n}$ and $\{ u_{1}, u_{2}, \; \cdots, \; u_{n} \}\;$ be that in second copy of $C_{n}$. According to the construction of the shadow graph $\; D_{2}(C_{n})$ of the cycle $C_{n}$, the edges set in the shadow graph $\; D_{2}(C_{n})$ are $\{ v_{i}v_{i+1}, \; v_{i}u_{i+1}, \; u_{i}v_{i+1}, \; u_{i}u_{i+1}\; , \; i = 1, 2, \; \cdots\; , n\; \}$. All the vertices $v_{i} \;$ and $u_{i}$ are of degree $4$, so the graph $\; D_{2}(C_{n})\;$ has $p = 2n$ vertices and $\; q = 4n$ edges, $k = max (p, q) = 4n$. There are two cases:

Case (1): When $\; n$ is odd, we define the labeling $f:E(D_{2}(C_{n})\longrightarrow \{\delta, \; 2 \delta, \; 3 \delta, \; \cdots, \; 4n\; \delta \}$ as follows:

In view of the above labeling pattern we have:

$\; \; \; f^{\ast}(v_{1}) = \; n\; \delta\; \; \;$ and $\; \; \; f^{\ast}(v_{i}) = [2\; \delta\; (n+i-1)] \; mod\; (4n)\; \delta, \; \; \; \; \; \; \; for\; \; \; \; 2 \leq i \leq n, \;$

$\; \; \; f^{\ast}(u_{n}) = \; 3\; n\; \delta\; \; \;$ and $\; \; \; f^{\ast}(u_{i}) = (2\; \delta\; i) \; mod\; (4n)\; \delta, \; \; \; \; \; \; \; for\; \; \; \; 1 \leq i \leq n-1. \;$

Hence the labels of the vertices $v_{2}, \; v_{3}, \; v_{4}, \; \cdots, \; v_{n-1}, \; v_{n}$ are $\; 2\; \delta(n+1), \; 2\; \delta(n+2), \; 2\; \delta(n+3), \; \cdots, \; 2\; \delta(2n-2), \; 2\; \delta(2n-1)$, respectively, and the labels of the vertices $u_{1}, \; u_{2}, \; u_{3}, \; \cdots, \; u_{n-1}, \; u_{n}$ are $\; 2\; \delta, 4\; \delta, \; 6\; \delta, \; \cdots, \; 2\; \delta(n-1), \; 3\; \delta \; n$, respectively, which are distinct numbers.

Case (2): When $n$ is even, we define the labeling $\; f\;$ as follows:

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; f(u_{i}\; v_{i+1}) = i\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; 1 \leq i \leq n, \;$

$\; \; \; \; \; \; f(v_{1}\; v_{n}) = \delta (4n-1), \; \; \; \; \; \; \; \; \; f(v_{i}\; v_{i+1}) = \delta\; (2n+i), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; 1 \leq i \leq n-1, \;$

$\; \; \; \; \; \; f(v_{n}\; u_{1}) = \delta \; 2n, \; \; \; \; \; \; \; \; \; \; \; \; f(v_{i}\; u_{i+1}) = \delta \; (4n-1-i), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; 1 \leq i \leq n-1, \;$

$\; \; \; \; \; \; f(u_{n}\; u_{1}) = \delta \; 4n, \; \; \; \; \; \; \; \; f(u_{i}\; u_{i+1}) = \delta \; (n+i), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; 1 \leq i \leq n-1.\;$

In view of the above labeling pattern we have:

$\; \; \; f^{\ast}(u_{1}) = (\; 3n+2)\; \delta, \; \; \; \; \; \; \; f^{\ast}(u_{n}) = (\; 2n-1)\; \delta, \; \; \; f^{\ast}(v_{1}) = \; (3n-2)\; \delta, \; \; \; \; \; \; f^{\ast}(v_{n}) = \; (2n-3)\; \delta, \; \; \; \;$

$\; \; \; \; \; \; \; f^{\ast}(u_{i}) = \; (2n-1+2i)\; \delta\; \; mod\; (4n)\; \delta, \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq i \leq n-1, \;$

$\; \; \; \; \; \; \; f^{\ast}(v_{i}) = [\; \delta\; (2i-3)] \; mod\; (4n)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; 2 \leq i \leq n-1.$

Obviously the vertex labels are all distinct. Thus, the graph $\; D_{2}(C_{n})\;$ is an edge $\delta-$ graceful graph for all $n$.

Illustration: The shadow graph $\; D_{2}(C_{9})\;$ with an edge $5-$ graceful labeling and $\; D_{2}(C_{10})\;$ with an edge $3-$ graceful labeling are shown in Figure 11.

3.3.   Edge $\delta-$ graceful labeling of the shadow graph $\; D_{2}(F_{n})$ of the fan graph $F_{n}$

Theorem 3.3. For any positive integer $\delta$, the shadow graph $\; D_{2}(F_{n}), \; \; n\geq 3\;$ of the fan graph $F_{n}$ is an edge $\; \delta-$ graceful graph.

Proof. Let $\{v_{0}, v_{1}, v_{2}, \; \cdots, \; v_{n} \}\;$ be the vertices in first copy of $F_{n}$ and $\{ u_{0}, u_{1}, u_{2}, \; \cdots, \; u_{n} \}\;$ be that in second copy of $F_{n}$. The edges set in the shadow graph $\; D_{2}(F_{n})$ are $\{ v_{i}u_{i+1}, \; v_{i}v_{i+1}, \; u_{i}v_{i+1}, \; u_{i}u_{i+1}, \; \; \; 1 \leq i \leq n-1 \}\cup \{\; u_{i}v_{0}\; , \; v_{i}u_{0}, \; v_{i}v_{0}\; , \; \; u_{0}u_{i}, \; \; \; 1 \leq i \leq n\; \}$, so the graph $\; D_{2}(F_{n})\;$ has number of total vertices $p = | V(D_{2}(F_{n})| = 2n+2$ and edges $q = | E(D_{2} (F_{n})| = 8n-4, \; \; n = 2, 3, 4, \; \cdots \; , \;$ see Figure 12. There are three cases:

Case (1): When $\; n\equiv 0 \; mod\; 4\;$ and $\; \; n\equiv 2 \; mod\; 4$, we define the labeling

$f:E[D_{2}(F_{n})]\longrightarrow \{\delta, \; 2 \delta, \; 3 \delta, \; \cdots, \; (8n-4)\; \delta \}$ as follows:

In view of the above labeling pattern, we can check that

(i) $[f (v_{0}\; v_{i})+ f (v_{0}\; u_{i })]\; \; mod\; [(8n-4)\; \delta\; ] = \; 0, \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 1\; \leq\; i \leq n.$

(ii) $[f (u_{0}\; u_{n-i})+ f (u_{0}\; v_{i+2})]\; mod\; [(8n-4)\; \delta] \; = 0, \; \; \; \; \; \; for\; \; \; \; 0\; \leq\; i \leq \; \frac{n}{2}$.

Then, the induced vertex labels are:

$f^{\ast}(v_{0}) = \sum_{i = 1}^{n} [\; f (v_{0}\; v_{i}) + \; f (v_{0}\; u_{i})]\; mod\; [(8n-4)\; \delta] = 0$,

$f^{\ast}(u_{0}) = \sum_{i = 1}^{n} [\; f (u_{0}\; v_{i}) + \; f (u_{0}\; u_{i})]\; mod\; [(8n-4)\; \delta] = \; [f (u_{0}\; v_{1})+\; f (u_{0}\; u_{1})] mod\; (8n-4)\; \delta = (4n-2) \delta$,

$f^{\ast}(v_{1}) = \; (n-1)\; \delta, \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{n}) = \; (5n)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{1}) = \; (4n-3)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{n}) = \; n\; \delta$,

$f^{\ast}(v_{i}) = [f (v_{i}\; v_{i+1})+ f (v_{i-1}\; v_{i})+f(v_{i}\; u_{0})+ f(v_{i}\; v_{0})+f(v_{i}\; u_{i+1})+ f(u_{i-1}\; v_{i})] \; mod\; [(8n-4)\; \delta]$

$\; \; \; \; \; \; \; \; \; = [\; \delta\; (8n-4i)] \; mod\; [(8n-4)\; \delta], \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\; \leq\; i \leq n -1$.

$f^{\ast}(u_{i}) = [f (u_{i}\; u_{i+1})+ f (u_{i-1}\; u_{i})+f(u_{i}\; u_{0})+ f(u_{i}\; v_{0})+f(u_{i}\; v_{i+1})+ f(v_{i-1}\; u_{i})] \; mod\; (8n-4)\; \delta$

$\; \; \; \; \; \; \; \; \; \; = [\; \delta\; (3n-2i)] \; mod\; [(8n-4)\; \delta], \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\; \leq\; i \leq n -1$.

Hence the labels of the vertices $v_{2}, \; v_{3}, \; v_{4}, \; \cdots, \; v_{n-2}, \; v_{n-1}$ are $\; \delta(8n-8), \; \delta(8n-12), \; \delta(8n-16), \; \cdots, \; \delta(4n+8), \; \delta(4n+4)$, respectively, and the labels of the vertices $u_{2}, \; u_{3}, \; u_{4}, \; \cdots, \; u_{n-2}, \; u_{n-1}$ are $\; \delta(3n-4), \; \delta(3n-6), \; \delta(3n-8), \; \cdots, \; \delta(n+4), \; \delta (n+2)$, respectively, which are distinct numbers.

Case (2): When $\; n\equiv 1 \; mod\; 4$, we define the labeling $f$ as follows:

In view of the above labeling pattern, the induced vertex labels are:

$\; \; f^{\ast}(v_{0}) = \; 0, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{1}) = 4\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{2}) = 8\; \delta, \;$

$\; \; f^{\ast}(v_{3}) = \; \delta\; (2n+7), \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{n-1}) = 5n\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{n}) = \; \delta$,

$\; \; f^{\ast}(u_{0}) = \; (4n-3)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{1}) = (7n-6)\; \delta\; , \; \; \; \; f^{\ast}(u_{n}) = (5n-3)\; \delta, \; \;$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{i}) = [\; \delta\; (2n+2+2i)] \; mod\; [(8n-4)\; \delta], \; \; \; \; \; \; \; \; for\; \; \; 4\; \leq\; i \leq n-2$.

Finally, $\; \; \; f^{\ast}(u_{i}) \; = [\; \delta\; (7n-4-4i)] \; mod\; [(8n-4)\; \delta], \; \; \; \; \; \; \; \; \; for\; \; \; 2\; \leq\; i \leq n-1$.

Hence the labels of the vertices $v_{2}, \; v_{3}, \; v_{4}, \; \cdots, \; v_{n-2}\; , v_{n-1}$ are $\; \delta(8n-8), \; \delta(8n-12), \; \delta(8n-16), \; \cdots, \; \delta(4n+8), \; \delta(4n+4)$, respectively, and the labels of the vertices $u_{2}, \; u_{3}, \; u_{4}, \; \cdots, \; u_{n-2}, \; u_{n-1}$ are $\; \delta(7n-12), \; \delta(7n-16), \; \delta(7n-20), \; \cdots, \; \delta(3n+4), \; \delta (3n)$, respectively, which are distinct numbers.

In this labeling, the induced labeling of the vertex $u_{0}$ will equal the induced labeling of the vertex

$\; u_{i}$ when $i = \dfrac{3n-1}{4}\; \;$ i.e., when $\; \; n = 4k+3\; \; \Rightarrow\; \; n\equiv 3 \; mod\; 4$.

Case (3): When $\; n\equiv 3 \; mod\; 4$, we define the labeling $f$ as in the case $\; n\equiv 1 \; mod\; 4\;$ but we change the labeling of two edges $(u_{0}v_{n})$ and $(u_{1}v_{2})$ as follows:

$\; \; \; \; f(u_{0}v_{n}) = \delta\; (4n-1)\; \; \; \; \;$ and $\; \; \; \; \; f(u_{1}v_{2}) = \delta\; (4n-3)$.

The induced vertex labels are:

$\; \; \; f^{\ast}(v_{0}) = \; 0, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{1}) = 4\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{2}) = 6\; \delta,$

$\; \; \; f^{\ast}(v_{3}) = \; \delta\; (2n+7), \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{n-1}) = 5n\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{n}) = 3\; \delta, \; \;$

$\; \; \; f^{\ast}(u_{0}) = \; (4n-1)\; \delta, \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{1}) = (7n-8)\; \delta\; , \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{n}) = (5n-3)\; \delta$,

$\; \; \; \; \; \; \; \; f^{\ast}(v_{i}) = [\; \delta\; (2n+2+2i)] \; mod\; [(8n-4)\; \delta], \; \; \; \; \; \; \; \; for\; \; \; \; \; 4\; \leq\; i \leq n-2 \;$

$\; \; \; \; \; \; \; \; f^{\ast}(u_{i}) \; = [\; \delta\; (7n-4-4i)] \; mod\; [(8n-4)\; \delta], \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\; \leq\; i \leq n-1$.

Hence the vertex labels are all distinct and a multiple of $\delta$. Therefore the shadow graph $\; D_{2} (F_{n})$ admits an edge $\; \delta-$ graceful labeling.

Illustration: The shadow graphs $\; D_{2} (F_{8})$ with edge $4-$ graceful labeling, $\; D_{2} (F_{9})$ with edge $3-$ graceful labeling and $\; D_{2} (F_{7})$ with an edge $5-$ graceful labeling. are presented in Figure 13.

4.   Edge $\; \delta-$ graceful labeling of the middle graph of some graphs

4.1.   Edge $\; \delta-$ graceful labeling of the middle graph $\; M(P_{n})\;$

Theorem 4.1. For any positive integer $\delta\;$, the middle graph $\; M(P_{n})\;$ of path $P_{n}$ is an edge $\; \delta-$ graceful graph when $\; n\;$ is even and $n\equiv\; 1 \; mod \; 4$.

Proof. Let $\{ v_{1}, \; v_{2}, \; \cdots, \; v_{n} \; \}$ be the vertices and $\{ u_{1}, \; u_{2}, \; \cdots, \; u_{n-1} \}\;$ be the edges of path $P_{n}$. Then $\; V [M(P_{n})] = V (P_{n}) \cup E(P_{n})\;$ and $\; E[M(P_{n})] = \{v_{i}u_{i}; \; 1 \leq i \leq n -1, \; \; v_{i}u_{i-1};\; 2 \leq i \; \leq n, \; u_{i}u_{i+1};\; 1 \leq i \leq n- 2 \}$. Here $p = \; 2n-1\;$ vertices and $\; q = 3n-4\;$ edges, $\; k = max (p, q)\; = \; 3n-4$.

Case (1): When $n \;$ is even, We define the labeling function $f: E(M(P_{n})) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; (3n -4)\delta \}$ as follows:

Then the induced vertex labels are:

$f^{\ast}(v_{1}) = \; \; \delta(n-3), \; \; \; \; \; \; \; \; f^{\ast}(v_{n}) = 0, \; \; \; \; \; \; f^{\ast}(u_{1}) = \; (n-5)\; \delta, \; \; \; \; \; \; \; f^{\ast}(u_{n-1}) = \; (3n-5)\; \delta, \;$

$f^{\ast}(u_{n-2}) = \; (3n-6)\; \delta, \; \; \; \; \; \; \; f^{\ast}(u_{\frac{n}{2}}) = \; (3n-7)\; \delta, \; \; \; \; and\; \; \; f^{\ast}(u_{\frac{n}{2}-1}) = \; (2n-2)\; \delta, \;$

$f^{\ast}(v_{i}) = [f(v_{i}u_{i-1})+f(v_{i}u_{i})] \; mod\; [(3n-4)\delta] = \; (2i-2)\; \delta, \; \; \; \; \; \; \; for \; \; \; \; 2 \leq i \leq n-1; \; \;$

$f^{\ast}(u_{i}) = [f(u_{i-i}u_{i})+f(u_{i}u_{i+1}) + f(u_{i}v_{i})+f(u_{i}v_{i+1})] \; mod\; [(3n-4)\delta]$

$\; \; \; \; \; \; \; \; \; \; = \; (2n-3-2i)\; \delta, \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; \; 2 \leq \; i\; \leq \frac{n}{2}-2 \; , \; \;$ and

$f^{\ast}(u_{i}) = \; (4n-9-2i)\; \delta, \; \; \; \; \; \; for\; \; \; \; \; \; \frac{n}{2}+1 \leq i \leq\; \cdots, n-3$.

Case (2): When $n\equiv\; 1 \; mod \; 4$, We define the labeling function $f: E(M(P_{n})) \longrightarrow \{\delta, \; 2\; \delta, \cdots, \; (3n -4)\delta \}$ as follows:

Then the induced vertex labels are:

$f^{\ast}(v_{1}) = \; \; \delta, \; \; \; \; \; \; \; \; f^{\ast}(v_{n}) = n\; \delta, \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{1}) = \; 5\; \delta, \; \; \; \; \; \; \; f^{\ast}(u_{n-1}) = \; (n+1)\; \delta, \;$

$f^{\ast}(v_{i}) = [f(v_{i}u_{i-1})+f(v_{i}u_{i})] \; mod\; [(3n-4)\delta] = \; (2n-3+2i)\; \delta mod\; [(3n-4)\delta]\; \; \; \; \; 2 \leq\; i\leq\; n-1, \; \;$

$f^{\ast}(u_{i}) = [f(u_{i-i}u_{i})+f(u_{i}u_{i+1}) + f(u_{i}v_{i})+f(u_{i}v_{i+1})] \; mod\; [(3n-4)\delta]$

$\; \; \; \; \; \; \; \; \; = \; [(n+1+4i)\; \delta]\; mod\; [(3n-4)\delta], \; \; \; \; \; \; \; for\; \; \; \; 2\; \leq i \leq n-2$.

Hence the labels of the vertices $\; v_{2}, \; v_{3}, \; \cdots, v_{n-2}, \; v_{n-1}\;$ are $\; (2n+1) \delta, \; (2n+3) \delta, \; \cdots, \; (n-3) \delta, \; (n-1)\delta$, respectively, and the labels of the vertices $\; u_{2}, \; u_{3}, \; \cdots, u_{n-3}, \; u_{n-2}\;$ will be $\; (n+9) \delta, \; (n+13) \delta, \; \cdots, \; (2n-7) \delta, \; (2n-3) \delta$, respectively. Hence the vertex labels are all distinct and a multiple of $\delta$. Therefore $M(P_{n})$ admits an edge $\delta$ graceful labeling when $n$ is odd.

Illustration: The middle graphs $\; M(P_{14})\;$ with an edge $2-$ graceful labeling and $\; M(P_{13})\;$ with an edge $3-$ graceful labeling are shown in Figure 14.

4.2.   The middle graph $\; M(C_{n}), \; \; n\geq 3\;$ of the cycle $C_{n}$

Theorem 4.2. For any positive integer $\delta$, the middle graph $\; M(C_{n}), \; \; n\geq 3\;$ of the cycle $C_{n}$ is an edge $\; \delta-$ graceful graph when $n$ is odd number.

Proof. Let $\{ v_{1}, v_{2}, \; \cdots, \; v_{n} \; \}$ be the vertices of the cycle $\; C_{n}$ and $\{ u_{1}, u_{2}, \; \cdots, \; u_{n-1} \; \}$ be the edges of the cycle $\; C_{n}$. The $\; V [M(C_{n})] = V (C_{n}) \cup E(C_{n})\;$ and $\; E[M(C_{n})] = \{u_{i}u_{i+1}, \; \; v_{i}u_{i}, \; \; u_{i}v_{i+1};\; \; 1 \leq i \leq n \}$, so the number of vertices $p = | V[M(C_{n}]| = 2n$ and edges $q = | E[M(C_{n})]| = 3n, \; \; n = 3, 4, \; \cdots \; .\;$

When $n$ is odd, we define the labeling function $f: E(M(C_{n})\; ) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; (3n)\delta \}$ as follows:

In view of the above labeling pattern, the induced vertex labels are:

$f^{\ast}(v_{1}) = \delta (3n-2), \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{i}) = \; (6i-8)\; \delta \; mod\; [(3n)\delta], \; \; \; \; \; \; \; \; \; \; \; \; \; \; 2\leq\; i \leq\; n, \; \;$

$\; f^{\ast}(u_{1}) = \; 2\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{i}) = \; (12i-10)\; \delta\; mod\; [(3n)\delta], \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; 2\leq\; i \leq\; n$.

Hence the vertex labels are all distinct and a multiple of $\; \delta$. Therefore $M(C_{n})\;$ admits an edge $\; \delta-$ graceful labeling.

Illustration: The middle graph $\; M(C_{9})\;$ of the cycle $C_{9}$ with an edge $6-$ graceful labeling is shown in Figure 15.

4.3.   The middle graph $\; M (Cr_{n})$ of the crown graph $\; Cr_{n}$

Theorem 4.3. For any positive integer $\delta$, the middle graph of the crown graph $\; Cr_{n}$ is an edge $\; \delta-$ graceful graph.

Proof. Let $\{ v_{1}, \; v_{2}, \; \cdots, v_{n} \}$ and $\{ u_{1}, \; u_{2}, \; \cdots, u_{n-1} \; \}$ be the vertices of the crown graph $\; Cr_{n}$ and $v^{\prime}_{1}, \; v^{\prime}_{2}, \; \cdots, \; v^{\prime}_{n} \;$ and $u^{\prime}_{1}, \; u^{\prime}_{2}, \; \cdots, \; u^{\prime}_{n-1} \;$ be the edges of the crown graph $\; Cr_{n}$. Then $\; V (M(Cr_{n})) = V (Cr_{n}) \cup E(Cr_{n})\;$ and $\; E(M(Cr_{n})) = \{v_{i}u^{\prime}_{i}, \; \; v^{\prime}_{i}v_{i+1}, \; \; v_{i}v^{\prime}_{i}, \; \; u_{i}u^{\prime}_{i}, \; \; u^{\prime}_{i}v^{\prime}_{i}, \; \; v^{'}_{i}u^{\prime}_{i+1}, \; \; \; v^{\prime}_{i}v^{\prime}_{i+1};\; \; 1 \leq i \leq n \}$, so $p = \; 4n\;$ vertices and $\; q = 7n\;$ edges, $k = max (p, q)\; = 7n$, see Figure 16.

Case (1): When $n$ is odd, we define the labeling function $f: E(M(Cr_{n})) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; 7n \delta \}$ as follows:

Then the induced vertex labels are:

$f^{\ast}(v^{\prime}_{1}) = \; (2n+1)\; \delta, \; \; \; \; \; \; f^{\ast}(v^{\prime}_{\frac{n+1}{2}}) = \; (2n+2)\; \delta, \; \; \; \; \; \; \; f^{\ast}(v^{\prime}_{\frac{n+3}{2}}) = \; (3n)\; \delta, \; \; \; \; \; \; f^{\ast}(v^{\prime}_{n}) = \; (3n+3)\; \delta, \; \; \; \; \;$

$f^{\ast}(v_{1}) = \; (3n+2)\; \delta, \; \; \; \; \; \; \; \; \; f^{\ast}(u_{1}) = \; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u^{\prime}_{1}) = \; (n+2)\; \delta,$

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{i}) = \; (4n+i+2)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; 2 \leq\; i\leq n$,

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{i}) = \; (7n-i+1)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq\; i\leq n$,

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u^{\prime}_{i}) = \; (2i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2 \leq\; i\leq n$,

Hence the labels of the vertices $\; v^{\prime}_{2}, \; v^{\prime}_{3}, \; \cdots\; , v^{\prime}_{\frac{n-3}{2}}, \; v^{\prime}_{\frac{n-1}{2}}\;$ are $\; (2n-1)\; \delta, \; (2n-3)\; \delta, \; \cdots, \; (n+6) \; \delta, \; (n+4)\; \delta$, respectively, and the labels of the vertices $\; v^{\prime}_{\frac{n+5}{2}}, \; v^{\prime}_{\frac{n+7}{2}}, \; \cdots\; , v^{\prime}_{n-2}, \; v^{\prime}_{n-1}\;$ are $\; (3n-2)\; \delta, \; (3n-4)\; \delta, \cdots, \; (2n+7) \; \delta, \; (2n+5)\; \delta$, respectively.

Case (2): When $\; n \equiv\; 0\; mod \; 6, \;$ and $\; n \equiv\; 2\; mod \; 6$. We define the labeling function $f: E(M(Cr_{n})) \longrightarrow \{\delta, \; 2\; \delta, \cdots, \; 7n \delta \}$ as follows:

Then the induced vertex labels are:

$f^{\ast}(v_{1}) = \; (3n+3)\; \delta, \; \; \; \; \; f^{\ast}(v_{i}) = \; (4n+i+2)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\; \leq i\leq n,$

$f^{\ast}(u_{1}) = \; (3n+1)\; \delta, \; \; \; \; \; f^{\ast}(u_{i}) = \; (7n-i+1)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\; \leq i\leq n,$

$f^{\ast}(u^{\prime}_{1}) = \; 3\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u^{\prime}_{i}) = \; (2i+1)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\; \leq i\leq n,$

$f^{\ast}(v^{\prime}_{1}) = \; (5n+4)\; \delta, \; \; \; \; \; \; \; f^{\ast}(v^{\prime}_{\frac{n}{2}+1}) = \; (2n+5)\; \delta, \; \; \; \; \; \; \; \; \; \; f^{\ast}(v^{\prime}_{n}) = \; (3n)\; \delta\; \; \; \; \;$ and

Hence the labels of the vertices $\; v^{\prime}_{2}, \; v^{\prime}_{3}, \; \cdots\; , \; v^{\prime}_{\frac{n}{2}-1}, \; v^{\prime}_{\frac{n}{2}}\;$ are $\; 10 \delta, \; 16\; \delta, \; \cdots, \; (3n-8) \; \delta, \; (3n-2)\; \delta$, respectively, and the labels of the vertices $\; v^{\prime}_{\frac{n}{2}+2}, \; v^{\prime}_{\frac{n}{2}+3}, \; \cdots\; , v^{\prime}_{n-2}, \; v^{\prime}_{n-1}\;$ are $\; (n+12)\; \delta, \; (n+18)\; \delta, \; \cdots, \; (4n-12) \; \delta, \; (4n-6)\; \delta$, respectively.

We notice that, these values are distinct for all $n$ even except when $\; n \equiv\; 4\; mod \; 6, \;$ since

$\; v^{\prime}_{\frac{n}{2}+2} = \; (n+12)\; \delta = \; v^{\prime}_{3} = \; 16\; \delta, \; \; \; \; \; \; \; when\; \; \; n = 4\;$

$\; v^{\prime}_{\frac{n}{2}+2} = \; (n+12)\; \delta = \; v^{\prime}_{4} = \; 22\; \delta, \; \; \; \; \; \; \; when\; \; \; n = 10\;$

$\; v^{\prime}_{\frac{n}{2}+2} = \; (n+12)\; \delta = \; v^{\prime}_{5} = \; 28\; \delta, \; \; \; \; \; \; \; when\; \; \; n = 16\;$

Case (3): When $\; n \equiv\; 4\; mod \; 6$. Define the labeling function $f: E(M(Cr_{n})) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; 7n \delta \}$ as in the first case with changes in the following edges

Then the induced vertex labels are:

$f^{\ast}(v^{\prime}_{1}) = \; (5n+4)\; \delta, \; \; \; \; \; \; \; f^{\ast}(v^{\prime}_{\frac{n}{2}+1}) = \; (2n+5)\; \delta, \; \; \; \; \; \; \; \; \; \; f^{\ast}(v^{\prime}_{n}) = \; (3n-2)\; \delta\; \; \; \; \;$ and

Hence the labels of the vertices $\; v^{\prime}_{2}, \; v^{\prime}_{3}, \; \cdots\; , \; v^{\prime}_{\frac{n}{2}-1}, \; v^{\prime}_{\frac{n}{2}}\;$ are $\; 12 \delta, \; 18\; \delta, \; \cdots, \; (3n-6) \; \delta, \; (3n)\; \delta$, respectively, and the labels of the vertices $\; v^{\prime}_{\frac{n}{2}+2}, \; v^{\prime}_{\frac{n}{2}+3}, \; \cdots\; , v^{\prime}_{n-2}, \; v^{\prime}_{n-1}\;$ are $\; (n+10)\; \delta, \; (n+16)\; \delta, \; \cdots, \; (4n-14) \; \delta, \; (4n-8)\; \delta$, respectively. Hence there are no repetition in the vertex labels which completes the proof.

Illustration: The middle graphs $\; M (Cr_{10})$ with an edge $3-$ graceful labeling and $\; M (Cr_{9})$ with an edge $2-$ graceful labeling are shown in Figure 17.

5.   Edge $\; \delta-$ graceful labeling of the total graph of some graphs

5.1.   Edge $\; \delta-$ graceful labeling of the total graph $\; T{(P_{n})}\;$

Theorem 5.1. For any positive integer $\delta$, the total graph $\; T{(P_{n})}\;$ of the path $P_{n}\;$ is an edge $\; \delta-$ graceful graph.

Proof. Let $\{ v_{1}, \; v_{1}, \; \cdots, \; v_{n} \}\;$ be the vertices of the path $P_{n}\;$, the total graph $\; T(P_{n})$ of path $P_{n}$ has vertices set $V(T(P_{n})) = \{ v_{i}, \; \; 1 \leq i \leq n \}\; \cup \{ u_{i}, \; 1 \leq i \leq n-1 \}$ and edges set $E (T(P_{n})) = \{ v_{i}v_{i+1}, \; v_{i}u_{i}, \; u_{i}v_{i+1}, \; \; 1 \leq i \leq n-1\}\; \cup \{ u_{i}u_{i+1}, \; \; 1 \leq i \leq n-2 \}$, so the graph $\; T(P_{n})\;$ has $p = \; 2n-1$ and $\; q = 4n-5$, $k = max (p, q)\; = \; 4n-5$.

If $\; n = 2\;$ the graph $T(P_{2})$ is an edge $\delta-$ graceful graph since it isomorphic to $\; C_{3}\;$ ^[15]. There are two cases:

Case (1): When $n$ is even, $n\geq 4$. We define the labeling function $f: E(T(P_{n})) \longrightarrow \{\delta, \; 2\delta, \; \cdots, \; (4n -5)\delta\; \}$ as follows:

In view of the above labeling pattern then the induced vertex labels are:

$f^{\ast}(v_{1}) = \; (n+1)\; \delta, \; \; \; f^{\ast}(v_{n}) = \; (n-1)\; \delta, \; \; \; f^{\ast}(u_{1}) = \; 0, \; \; \; f^{\ast}(u_{2}) = \; (2n-1)\; \delta, \; \; \; f^{\ast}(u_{n-1}) = (3n-5)\; \delta$,

$f^{\ast}(v_{i}) = \; [(2n-2+2i)\; \delta]\; mod\; [(4n-5)\delta], \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; 2\leq\; i\leq n-1,$

$f^{\ast}(u_{i}) = \; (2i-2)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; 3\leq\; i\leq n-2.$

Hence the labels of the vertices are all distinct numbers.

Case (2): When $\; n$ is odd:

$\; \bullet\;$ If $\; n = 3\;$ the graph $T(P_{3})$ is an edge $\delta-$ graceful graph for any positive integer $\delta$ define the labeling function $f: E(T(P_{3})) \longrightarrow \{\delta, \; 2\delta, \; \cdots, \; 7\delta \}$ as shown in Figure 18.

$\; \bullet\;$ If $\; n \geq 5$. Define the labeling function $f: E(T(P_{n})) \longrightarrow \{\delta, \; 2\delta, \; \cdots, \; (4n -5)\delta \; \}$ as follows:

In view of the above labeling pattern then the induced vertex labels are:

$\; \; \; f^{\ast}(v_{1}) = \; \delta, \; \; \; \; \; \; f^{\ast}(v_{2}) = \; (n+3)\delta, \; \; \; \; \; \; \; f^{\ast}(v_{n-1}) = \; (n+1)\; \delta,$

$\; \; \; f^{\ast}(v_{n}) = \; 2\; \delta, \; \; \; \; \; \; f^{\ast}(u_{1}) = \; 3\; \delta, \; \; \; \; \; \; \; f^{\ast}(u_{n-1}) = \; n\; \delta,$

$f^{\ast}(v_{i}) = [(1+2i)\; \delta]\; mod\; [(4n-5)\delta], \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 3\leq\; i\leq n-2, \; \;$ and

$f^{\ast}(u_{i}) = [\; (2n-1+2i)\; \delta]\; mod\; [(4n-5)\delta], \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; 2\leq\; i\leq n-2.$

Hence the labels of the vertices $\; v_{3}, \; v_{4}, \; \cdots, v_{n-3}, \; v_{n-2}\;$ will be $\; 7\; \delta, \; 9\; \delta, \; \cdots, \; (2n-5) \; \delta, \; (2n-3) \; \delta$, respectively, and the labels of the vertices $\; u_{2}, \; u_{3}, \; \cdots, \; u_{n-3}, \; u_{n-2}\;$ will be $\; (2n+3)\; \delta, \; (2n+5)\; \delta, \; \cdots, \; (4n-7)\; \delta, \; 0$, respectively.

In this case we have $f^{\ast}(v_{\frac{n-1}{2}})$ will equal $f^{\ast}(u_{n-1})$, so we change the labeling of two edges $(v_{\frac{n-5}{2}}v_{\frac{n-3}{2}})$ and $(v_{\frac{n-3}{2}}\; v_{\frac{n-1}{2}}\; )$ as follows $\; f(v_{\frac{n-5}{2}}v_{\frac{n-3}{2}}) = \dfrac{\delta}{2}(7n-11)$. and $f(v_{\frac{n-3}{2}}\; v_{\frac{n-1}{2}}\; ) = \dfrac{\delta}{2}(7n-13)$.

Then $f^{\ast}(v_{\frac{n-5}{2}}) = \; (n-3)\delta\;$ and $f^{\ast}v_{\frac{n-1}{2}} = (n-1)\; \delta$ Hence the vertex labels are all distinct and a multiple of $\; \delta$. Therefore $T(P_{n})\;$ admits an edge $\; \delta-$ graceful labeling.

Illustration: The total graphs $\; T(P_{10})\;$ and $\; T(P_{11})\;$ with an edge $6-$ graceful labeling are shown in Figure 19.

5.2.   Edge $\delta-$ graceful labeling of the total graph $\; T(C_{n})$

Theorem 5.2. For any positive integer $\delta$, the total graph $\; T(C_{n}), \; \; n\geq 3\;$ of the cycle $C_{n}$ is an edge $\; \delta-$ graceful graph.

Proof. Let $\{ v_{i}, \; \; i = 1, 2, \; \cdots\; , \; n \}\;$ be the vertices of the path $C_{n}\;$, the total graph $\; T(C_{n})$ of path $C_{n}$ has vertices set $V(T(C_{n})) = \{ v_{i}, \; u_{i}\; , \; i = 1, 2, \; \cdots\; , n \}$ and $E (T(C_{n})) = \{ v_{i}v_{i+1}, \; v_{i}u_{i}, \; u_{i}v_{i+1}, \; u_{i}u_{i+1}, \; i = 1, 2, \; \cdots\; , \; n \}$, so the graph $\; T(C_{n})\;$ has $p = \; 2n$ and $\; q = 4n$, $k = max (p, q) = 4n$. There are two cases:

Case (1): When $\; n$ is even, we define the labeling $f:E(T(C_{n})\longrightarrow \{\delta, \; 2\; \delta, \; 3\; \delta, \; \cdots, \; 4n\; \delta \}$ as follows:

In view of the above labeling pattern we have:

$\; \; \; f^{\ast}(v_{1}) = \; 0, \; \; \; \; \; f^{\ast}(u_{1}) = \; (2\; n\; +1)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{n}) = \; (\; n\; -1)\; \delta$,

$\; \; \; f^{\ast}(v_{i}) = 2\; \delta\; (i-1), \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; 2\leq\; i\leq n, \;$

$\; \; \; f^{\ast}(u_{i}) = \; \delta\; (2i+2n-1), \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; 2\leq\; i\leq n-1.\;$

Hence the labels of the vertices $v_{2}, \; v_{3}, \; v_{4}, \; \cdots, \; v_{n-1}\; , v_{n}$ are $\; 2\; \delta, \; 4\; \delta, \; 6\; \delta, \; \cdots, \; 2\; \delta(n-2), \; 2\; \delta(n-1)$, respectively, and the labels of the vertices $u_{2}, \; u_{3}, \; u_{4}, \; \cdots, \; u_{n-2}, \; u_{n-1}$ are $\; \delta\; (2\; n+3), \; \delta\; (2\; n+5), \; \delta\; (2\; n+7), \; \cdots, \; \delta(4n-5), \; \delta \; (4n-3)$, respectively.

It is easy to see that: $\; f^{\ast}(u_{n}) = \; f^{\ast}(v_{i})\; \;$ when $i = \dfrac{n+1}{2}, \;$ but $\; n \;$ is even number, so all the labels are distinct numbers.

Case (2): When $\; n$ is odd, we define the labeling $\; f\;$ as in the above case where $\; n$ is even but we change the labeling of two edges $(u_{n}v_{1})$ and $(v_{n}v_{1})$ as follows $\; f(u_{n}v_{1}) = \; 2\; n\; \delta$ and $f(v_{n}v_{1}) = n\; \delta$.

The induced vertex labels are:

$\; \; \; f^{\ast}(v_{1}) = [f (v_{1}\; v_{2})+ f (v_{n}\; v_{1})+ f(u_{1}\; v_{1})+ f(u_{n}\; v_{1})] \; mod\; (4n)\; \delta = \; 0, \; \;$

$\; \; \; f^{\ast}(v_{n}) = [f (v_{n}\; v_{1})+ f (v_{n-1}\; v_{n})+ f(u_{n}\; v_{n})+ f(u_{n-1}\; v_{n})] \; mod\; (4n)\; \delta = \; \delta\; \; (n-2), \;$

$\; \; \; f^{\ast}(u_{n}) = [f (u_{n}\; u_{1})+ f (u_{n-1}\; u_{n})+ f(u_{n}\; v_{1})+ f(u_{n}\; v_{n})] \; mod\; (4n)\; \delta = \; (\; 2n\; -1)\; \delta$.

It is easy to see that: $\; f^{\ast}(v_{n}) = \; f^{\ast}(v_{i})\; \;$ when $i = \dfrac{n}{2}$, but $n \;$ is odd number, so all the labels are distinct numbers.

Illustration: The total graph $\; T(C_{10})\;$ of the cycle $C_{10}$ with an edge $5-$ graceful labeling and the total graph $\; T(C_{9})\;$ of the cycle $C_{9}$ with an edge $6-$ graceful labeling are shown in Figure 20.

5.3.   Edge $\; \delta-$ graceful labeling of the total graph $\; T(Cr_{n})\;$

Theorem 5.3. For any positive integer $\delta$, the total graph of the crown graph $Cr_{n}$ is an edge $\; \delta-$ graceful graph.

Proof. Let $\{ v_{1}, \; v_{2}, \; \cdots, \; v_{n} \; \}$ and $\{ u_{1}, \; u_{2}, \; \cdots, \; u_{n} \; \}$ be the vertices of the crown graph $\; Cr_{n}$ and $\{ v^{\prime}_{1}, \; v^{\prime}_{2}, \; \cdots, \; v^{\prime}_{n} \}\;$ and $\{ u^{\prime}_{1}, \; u^{\prime}_{2}, \; \cdots, \; u^{\prime}_{n} \}\;$ be the edges of the crown graph $\; Cr_{n}$. Then $\; V (T(Cr_{n})) = V (Cr_{n}) \cup E(Cr_{n})\;$ and $\; E(T(Cr_{n})) = \{v_{i}u^{\prime}_{i}, \; v^{\prime}_{i}v_{i+1}, \; v_{i}v^{\prime}_{i}, \; \; u_{i}u^{\prime}_{i}, \; u^{\prime}_{i}v^{\prime}_{i}, \; v^{'}_{i}u^{\prime}_{i+1}, \; v^{\prime}_{i}v^{\prime}_{i+1}, \; \; v_{i}u_{i}, \; v_{i}v_{i+1};\; 1 \leq i \leq n \}\;$, see Figure 21.

Here $p = \; 4n\;$ vertices and $\; q = 9n\;$ edges, $k = max (p, q)\; = 9n$.

Case (1): When $\; n \;$ is odd. Define the labeling function $f: E(T(Cr_{n})) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; 9n\; \delta \}$ as follows:

Then the induced vertex labels are:

$f^{\ast}(v_{1}) = \; (8n+3)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{i}) = \; (8n+4i-1)\; \delta \; \; mod\; [(9n)\delta], \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\leq\; i\leq n$,

$f^{\ast}(u^{\prime}_{1}) = \; (5n+3) \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u^{\prime}_{i}) = \; (2n+2i+1)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\leq\; i\leq n$,

$\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u_{i}) = \; (2n+3-2i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 1\leq\; i\leq n$,

$f^{\ast}(v^{\prime}_{1}) = \; (n+3)\; \delta, \; \; \; \; \; \; \; f^{\ast}(v^{\prime}_{n}) = \; (6n+1)\; \delta, \; \; \; \; \; \; \; \; f\; (v^{\prime}_{i}) = (3n+1+2i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\leq\; i\leq n-1$.

Case (2): When $\; n\; \equiv\; 2\; mod \; 4$. The labeling function $f: E(T(Cr_{n})) \longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; 9n\; \delta \}$ defined as follows:

Then the induced vertex labels are:

$\; f^{\ast}(u_{i}) = \; (8n+2i)\; \delta \; mod\; [(9n)\delta], \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; 1\leq\; i\leq n$,

$f^{\ast}(u^{\prime}_{1}) = \; (2n+3) \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u^{\prime}_{i}) = \; (n+4i-1)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; 2\leq\; i\leq n$,

$\; \; f^{\ast}(v_{i}) = \; (8n+4i+1)\; \delta \; \; mod\; [(9n)\delta], \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 1\leq\; i\leq n$,

$f^{\ast}(v^{\prime}_{1}) = \; (3n+6)\; \delta, \; \; \; \; \; \; \; f^{\ast}(v^{\prime}_{n}) = \; (6n+2)\; \delta, \; \; \; \; f\; (v^{\prime}_{i}) = (3n+2+4i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; 2\leq\; i\leq n-1$.

These values are distinct for all $n \equiv\; 2\; mod \; 4$, but

$\; f^{\ast} (v^{\prime}_{i}) = \; (3n+4i+2)\; \delta = \; f^{\ast}(v^{\prime}_{n}) = \; (6n+3)\; \delta, \; \;$ when $\; \; i = \; \dfrac{3n}{4}\;$ i.e., when $\; \; n \equiv\; 0\; mod \; 4.\;$

Case (3): When $\; n\; \equiv\; 0\; mod \; 4\;$ define the labeling function $\; f\;$ as follows:

Then the induced vertex labels are:

$f^{\ast}(u_{i}) = \; (8n+2i)\; \delta \; mod\; [(9n)\delta], \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; \; 1\leq\; i\leq n$,

$f^{\ast}(u^{\prime}_{1}) = \; (4n+2) \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f^{\ast}(u^{\prime}_{i}) = \; (n+4i-1)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\leq\; i\leq n$,

$f^{\ast}(v^{\prime}_{1}) = \; (6n+3)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; f\; (v^{\prime}_{i}) = (3n+4i)\; \delta, \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\leq\; i\leq n$,

$f^{\ast}(v_{1}) = \; (6n+6)\; \delta, \; \; \; \; \; \; f^{\ast}(v_{n}) = \; (n+2)\; \delta$,

$f^{\ast}(v_{i}) = \; (8n+4i+1)\; \delta \; \; mod\; [(9n)\delta], \; \; \; \; \; \; \; \; \; \; \; for\; \; \; \; 2\leq\; i\leq n-1$.

We notice that, these values are distinct for all $n \equiv\; 0\; mod \; 4$. It should be notice that:

$\; f^{\ast}(v^{\prime}_{i}) = \; (3n+4i)\; \delta = \; f^{\ast} (u^{\prime}_{1}) = \; (4n+2)\; \delta, \; \; \; \;$ when $\; \; i = \; \dfrac{n+2}{4}\;$ i.e., when $\; n \equiv\; 2\; mod \; 4$. Hence there are no repetition in the vertex labels which completes the proof.

Illustration: The total graphs $T (Cr_{8})$ and $T (Cr_{9})$ of the crown graph with an edge $3-$ graceful labeling are shown in Figure 22.

6.   Edge $\; \delta-$ graceful labeling of the twig graph $\; TW_{_{n}}\;$

Definition 6.1. The twig graph $TW_{_{n}}, n\geq 4$ is the graph obtained from path $\; P_{n}\; = \{ v_{1}, v_{2}, \cdots, v_{n} \}\;$ by attaching exactly two pendant edges to each internal vertex of the path $P_{n}$.

Theorem 6.1. For any positive integer $\delta$, the twig graph $\; TW_{_{n}}\;$ is an edge $\; \delta-$ graceful graph when $\; n$ is odd.

Proof. Let $\{ v_{1}, v_{2}, \; \cdots, \; v_{n} \}\;$ be the vertices of the path $\; P_{n}\;$ and the new attaching vertices are $\{ u_{2}, u_{3}, \; \cdots, \; u_{n-1} \}\;$ and $\{ w_{2}, w_{3}, \; \cdots, \; w_{n-1} \}$. The edges of $\; TW_{n}\;$ are denoted by $\{ a_{i}, b_{i}, c_{i}, d_{i}\; \}$, where $a_{i} = \{v_{(\frac{n-4i+3}{2})}\; v_{(\frac{n-4i+5}{2})}\}$, $b_{i} = \{v_{(\frac{n+4i-3}{2})}\; v_{(\frac{n+4i-1}{2})}\}$, $d_{i} = \{v_{(\frac{n-4i +1}{2})}\; v_{(\frac{n-4i+3}{2})}\}$ and $c_{i} = \{ v_{(\frac{n+4i-1}{2})}\; v_{(\frac{n+4i+1}{2})}\}$ where,

The graph $\; TW_{_{n}}\;$ has $p = \; 3n-4$ vertices and $\; q = 3n-5$ edges, $k = max (p, q)\; = \; 3n-4$. see, Figure 23 We define the labeling function $f: E(TW_{n}) \longrightarrow \{\delta, 2 \delta, \cdots, (3n -4) \delta \}$ as follows:

At the beginning, we determine the middle vertex $v_{\frac{n+1}{2}}$ in the path $P_{n}$ and we start the labeling from this vertex by using the following algorithm.

First step: Label the edges $a_{1}, b_{1}, c_{1}$ and $d_{1}$ in the following order:

$f(a_{1}) = \delta$, $f(b_{1}) = \delta (3n-5)$, $f(c_{1}) = 2 \delta$ and

$f(d_{1}) = [f(b_{1})-f(a_{1})] mod [\delta (3n-4)] = \delta (3n-6).$

Second step: Label the vertices $a_{i}, b_{i}, c_{i}$ and $d_{i}$ by the following algorithm respectively

(i) $f(a_{i}) = [f(c_{i-1})-f(d_{i-1})] mod [\delta (3n-4)], \ \ for \ \ i = 2, 3, \cdots, \frac{n-3}{4}$,

(ii) $f(b_{i}) = (3n-4) \delta - f(a_{i})$,

(iii) $f(c_{i}) = [f(a_{i})-f(b_{i})] mod [\delta (3n-4)], \ \ for \ \ i = 2, 3, \cdots, \frac{n-7}{4}$,

(iv) $f(d_{i}) = [f(b_{i})-f(a_{i})] mod [\delta (3n-4)]. \ \ for \ \ i = 2, 3, \cdots, \frac{n-7}{4}$,

and repeats the second step until we fish the labeling of all edges in the path $P_{n}$.

At the end, the edges $(v_{i}u_{i})$ and $(v_{i}w_{i}), 2 \leq i \leq n-1$ take any number from the reminder set of the labeling such that $[f (v_{i}u_{i})+ f (v_{i}w_{i})] mod [\delta (3n-4)] \equiv 0 mod [\delta (3n-4)] for i = 2 \leq i \leq n-1$.

In view of the above labeling algorithm, the labels of the vertices are,

$f^{\ast}(v_{\dfrac{n+1}{2}}) = [f(a_{1})+ f(b_{1})+f (u_{ \dfrac{n+1}{2}})+ f (w_{\dfrac{n+1}{2}})] mod [\delta (3n-4)] = 0$,

$f^{\ast}(v_{\dfrac{n-1}{2}}) = f(b_{1}) = \delta (3n-5), f^{\ast}(v_{\dfrac{n+3}{2}}) = f(a_{1}) = \delta$,

$f^{\ast}(v_{\dfrac{n-3}{2}}) = f(c_{1}) = 2\delta, f^{\ast}(v_{\dfrac{n+5}{2}}) = f(d_{1}) = \delta (3n-6)$,

$f^{\ast}(v_{\dfrac{n-5}{2}}) = f(b_{2}) = \delta (3n-8), f^{\ast}(v_{\dfrac{n+7}{2}}) = f(a_{2}) = 4 \delta$,

$f^{\ast}(v_{\dfrac{n+(4i-1)}{2}}) = f(a_{i}), f^{\ast}(v_{\dfrac{n-(4i-3)}{2}}) = f(b_{i}), \ \ for \ \ i = 1, 2, \cdots, \frac{n-1}{4}$,

$f^{\ast}(v_{\dfrac{n+(4i+1)}{2}}) = f(d_{i}), f^{\ast}(v_{\dfrac{n-(4i-1)}{2}}) = f(c_{i}), \ \ for \ \ i = 1, 2, \cdots, \frac{n-5}{4}$.

The pendant vertices $v_{1} and v_{n}$ of the path $P_{n}$ will take the labels of its pendant edges, i.e.,

$\bullet$ If $n\equiv 1 (mod 4)$, then $f^{\ast}(v_{1}) = f(d_{\frac{n-1}{4}}), \ \ and \ \ \ f^{\ast}(v_{2n-1}) = f(c_{\frac{n-1}{4}})$.

$\bullet$ If $n\equiv 3 (mod 4)$, then $f^{\ast}(v_{1}) = f(a_{\frac{n+1}{4}}), \ \ and \ \ \ f^{\ast}(v_{n}) = f(b_{\frac{n+1}{4}})$.

It is clear that the labels of the vertices of the path $P_{n}$ take the labels of the edges of the path and each pendant vertex takes the labels of its incident edge. Then there are no repeated vertex labels, which completes the proof.

Illustration: The twig graph $TW_{13}$ is presented in Figure 24 with an edge $4-$ graceful labeling.

7.   Edge $\delta-$ graceful labeling of the snail graph $S_{n}$

Definition 7.1. A snail graph $S_{n}$ is obtained from path $P_{n} = \{ v_{1}, v_{2}, \cdots, v_{n}\}$ by attaching two parallel edges between $v_{i}$ and $v_{n-i+1}$ for $i = 1, 2, \cdots, \lfloor \frac{n}{2} \rfloor$.

Theorem 7.1. For any positive integer $\delta$, the snail graph $S_{n}$, $n > 2$ is an edge $\delta-$ graceful graph.

Proof.

Case (1): When $n$ is even, the vertices of the graph $S_{n}$ are $\{ v_{0}, v_{1}, v_{2}, \cdots, v_{n} \}$ and the edges are $\{ e_{1}, e_{1}, \cdots, e_{n-1}, a_{1}, a_{2} \cdots, a_{ \frac{n}{2}}, b_{1}, b_{2} \cdots, b_{ \frac{n}{2}}\}$ where $\{ e_{i} = v_{i}v_{i+1}, i = 1, 2, \cdots, n-1 \}$ and $\{ a_{j} = b_{j} = v_{i}v_{n-i+1}, j = 1, 2, \cdots, \frac{n}{2} \}$, see Figure 25, in this case $p = n, q = 2n-1$ and $k = max (p, q) = 2n-1$.

We define the labeling function $f: E(S_{n}))\longrightarrow \{\delta, \; 2\; \delta, \; \cdots, \; (2n-1)\; \delta \}$ as follows:

The induced vertex labels are:

$f^{\ast}(v_{1}) = [\; f(a_{1})\; + f(b_{1})\; + f(e_{1})\; ] \; mod\; [(2n-1)\; \delta] = \; \delta[\frac{n}{2}+1]$,

$f^{\ast}(v_{n}) = [\; f(a_{1})\; + f(b_{1})\; + f(e_{n-1})\; ] \; mod\; [(2n\; -1)\delta] = \; \delta\; [\frac{3n}{2}-2]$,

$\; \; \; \; \; \; f^{\ast}(v_{\frac{n}{2}}) = \delta\; (n-1), \; \; \; \; \; \; \; \; \; \; f^{\ast}(v_{\frac{n}{2}+1}) = \delta\; n$.

$\; f^{\ast}(v_{i}) = [\; f(a_{i})\; + f(b_{i})\; + f(e_{i})\; + f(e_{i-1})\; ] \; [mod\; (2n-1)\; \delta]$

$\; \; \; \; \; \; \; \; \; \; = \begin{cases} \delta\; (n+2i-1) \; mod\; [(2n-1)\; \delta]\; \; \; & \text{if }\; \; \; 2 \leq i \leq \frac{n}{2}-1;\\ \delta\; (n+ 2i-3) \; mod\; [(2n-1)\; \delta] \; \; \; & \text{if }\; \; \; \; \; \; \frac{n}{2}+2 \leq i \leq n-1. \\ \end{cases}$

Hence the labels of the vertices $v_{2}, \; v_{3}, \; \cdots, \; v_{\frac{n}{2}-1}\;$ are $\; \delta\; (n+3), \; \delta\; (n+5), \; \cdots, \; \delta\; (2n-3)$, respectively, and the labels of the vertices $\; v_{\frac{n}{2}+2}, \; v_{\frac{n}{2}+3}, \; \cdots, \; v_{n-1}$ are $\; 2\; \delta, \; 4\; \delta, \; \cdots, \; \delta\; (n-4)$, respectively.

Note that $S_{4}$ is an edge $\delta-$ graceful graph but not follow this rule, see Figure 26.

Case (2): When $\; n$ is odd. Let $\{ e_{1}, e_{1}, \; \cdots, \; e_{n-1}, a_{1}, a_{2}, \; \cdots, \; a_{ \frac{n-1}{2}}, b_{1}, b_{2}\; \cdots, \; b_{ \frac{n-1}{2}}\}\;$ be the edges of $\; S_{n}\;$ which are denoted as in the Figure 27, in this case $\; p = n, \; q = 2n-2, \; k = max (p, q)\; = \; 2n-2.\;$

Define the labeling function $f: E(S_{n}))\longrightarrow \{\delta, \; 2\; \delta, \cdots, (2n -2)\; \delta \}$ as follows:

Therefor, the induced vertex labels are

$\; \; f^{\ast}(v_{1}) = \; \delta \; (n+1)\; , \; \; \; \; \; \; f^{\ast}(v_{\frac{n+1}{2}}) = \; n \; \delta, \; \; \; \; \; \; \; \; f^{\ast}(v_{n}) = \; \delta\; \; \;$ and

$\; f^{\ast}(v_{i}) = [\; f(a_{ i})\; + f(b_{ i})\; + f(e_{i})\; + f(e_{i-1})\; ] \; mod\; [(2n-2)\delta]$

$\; \; \; \; \; \; \; \; \; \; = \delta\; (n+2i-1)\; mod\; [(2n-2)\delta], \; \; \; \; \; if\; \; \; i = 2, 3, \; \cdots\; , \frac{n-1}{2}, \frac{n+2}{2}, \; \cdots, n-1.$

Hence the labels of the vertices $v_{2}, v_{3}, \; \cdots, \; v_{\frac{n-3}{2}}\; , v_{\frac{n-1}{2}}\;$ are $\; \delta(n+3), \; \delta\; (n+5), \; \cdots, \; \delta\; (2n-4)\; , \; 0$, respectively, and the labels of the vertices $\; v_{\frac{n+3}{2}}, \; v_{\frac{n+5}{2} }, \; \cdots, \; v_{n-2}, \; v_{n-1}$ are $\; 4\; \delta, \; 6\; \delta, \; \cdots, (n-3)\; \delta, \; (n-1)\; \delta$, respectively. Overall all vertex labels are distinct and a multiple of $\delta\;$ numbers. Hence, the snail graph $\; S_{n}\;$ is an edge $\; \delta-$ graceful for all $\; n$.

Illustration: In Figure 28, we present a triple graceful labeling of the graph $S_{12}$ and an edge $4\;$ graceful labeling of $\; S_{13}$.

8.   Conclusions

Recently, edge graceful labeling of graphs has been studied too much and these objects continue to be an attraction in the field of graph theory and discrete mathematics. A senior number of published papers and results exist. So far, numerous graphs are unknown if it is edge graceful or not.

In this work, we pushed the new type of labeling (edge $\; \delta-$ graceful labeling), by finding more graphs that have edge $\; \delta-$ graceful labeling. We prove the existence of an edge $\; \delta-$ graceful labeling, for any positive integer $\; \delta$, for the following graphs. The splitting graphs of the cycle, fan, and crown graphs. The shadow graphs of the path, cycle, and fan graphs. The middle graphs and the total graphs of the path, cycle, and crown graphs. Finally, we display the existence of an edge $\; \delta-$ graceful labeling, for the twig and snail graphs.

Conflict of interest

The authors declare that they have no competing interests.