Enumeration of the Gutman and Schultz indices in the random polygonal chains

1.   Introduction

In this paper, we only consider simple and finite connected graphs. Chemical graph theory is a branch of mathematical chemistry which deals with the nontrivial applications of graph theory to solve molecular problems. The basic method is to model the molecular structure of a compound ^[1,2,3]. Each atom is represented by a vertex, and the chemical bonds between atoms are represented by the edges between the vertices. Thus, the entire molecular structure is represented by a diagram, which is called a molecular diagram. For more detailed information, we can refer to ^[4,5] and the references cited therein.

In this paper, chemical diagrams are studied by topological indices ^[6,7,8,9]. According to the different parameters such as point degree, adjacent point degree and distance between two points, topological indices can be divided into many categories. A graph G is an ordered $(V(G), E(G))$ consisting of a nonempty set V(G) of vertices, a set $E(G)$, disjoint from $V(G)$, of edges. The degree $d_{G}(v)$ (or $d(v)$ for short) of a vertex $v$ in $G$ is the number of edges of $G$ incident with $v$. The shortest distance between vertex $u$ and vertex $v$ is denoted by $d(u, v)$ ^{[10,11,12,13]}.

Wiener index is defined as ^[14]

The Wiener index is more and more widely used and studied, see ^[15,16,17]. Zhang, Li and so on ^[18,19] give the explicit analytical expressions for the expected values of the Schultz index, Gutman index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random polyphenylene chain. Now we will consider the random polygonal chains that are meaningful.

Gutman index is defined as

Schultz index is defined as

More articles on developing such a topology indices of the ^{[20,21,22,23]}, such as mathematical properties, discrimination and applications refer to ^{[24,25,26,27]}.

A random polygonal chain $G_{n}$ with $n$ polygons is made up of a polygonal chain $G_{n-1}$ with $n-1$ polygons to which a new terminal polygon $H_{n}$ by a cut edge, see Figure 1. When $n \geq 3$, the terminal polygon $H_{n}$ has $k$ connection ways, these connections are recorded as $G_{n}^{1}$, $G_{n}^{2}$, $G_{n}^{3}$, $\ldots$, $G_{n}^{k}.$ see Figure 2. A random polygonal chain $G_{n}(p_{1}, p_{2}, p_{3}, \ldots, p_{k-1})$ has $n$ polygons is a polygonal chain acquired by gradyally adding terminal polygons. Each step of adding can be randomly selected from $k$ connection methods:

● $G_{k-1}\rightarrow G_{2k}^{1}$ with probability $p_{1}$,

● $G_{k-1}\rightarrow G_{2k}^{2}$ with probability $p_{2}$,

● $\vdots$      $\vdots$          $\vdots$

● $G_{k-1}\rightarrow G_{2k}^{3}$ with probability $p_{3}$,

● $G_{k-1}\rightarrow G_{2k}^{k-1}$ with probability $p_{k-1}$,

● $G_{k-1}\rightarrow G_{2k}^{k}$ with probability $p_{k} = 1-p_{1}-p_{2}-p_{3}-\cdots-p_{k-1}$,

where the probabilities $p_{1}$, $p_{2}$, $p_{3}$, $\ldots$, $p_{k-1}$ are constants, there are independent of $k$.

Let $G_{n}$ be a polygonal chain with $n$ polygons $H_{1}$, $H_{2}$, $\ldots$, $H_{n}$. $u_{k}\omega_{k}$ is connecting $H_{k}$ and $H_{k+1}$ with $u_{k}\in V_{H_{k}}$ in $G_{n}$, $\omega_{k}\in V_{H_{k+1}}$ ($k = 1, 2, \ldots, n-1$). Obviously, both $\omega_{k }$ and $u_{k+1}$ are thevertices in $H_{k+1}$ and $d(\omega_{k}, u_{k+1}) \in \{1, 2, 3, \ldots, n\}$. Specially, $G_{n}$ is the meta-chain $M_{n}$, the ortho-chain $O_{n}^{1}$, $O_{n}^{2}$, $\ldots$, $O_{n}^{k-2}$and the para-chain $Ln$ if $d(\omega_{k}, u_{k+1}) = 1$ (i.e., $p_{1} = 1$), $d(\omega_{k}, u_{k+1})$ = 2 (i.e., $p_{2} = 1$), $d(\omega_{k}, u_{k+1})$ = 3 (i.e., $p_{3} = 1$), $\ldots$, $d(\omega_{k}, u_{k+1})$ = k (i.e., $p_{k} = 1$) ($\forall$ $k \in \{1, 2, \ldots, n-2\}$), respectively.

Zhang and Li et al.^[18], obtained the random polyphenylene chain expected values of some topological indices. We calculate the explicit analytical expressions for the expected values of the Gutman index, Schultz index of a random polygonal chain. Based on above results, we get the extremal values and average values of Gunman and Schultz indices of random polygonal chains.

2.   The Gutman index in a random polygonal chain

In this section, we will consider the expected values of Gutman index of the random polygonal chain. In fact, $G_{n+1}$ is $G_{n}$ linked to a new terminal polygonal $H_{n+1}$ by an edge, $H_{n+1}$ is made up with vertices $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{2k}$, and the new edge is $u_{n}x_{1}$; see Figure 1. For $\forall v \in V_{G_{n}}$,

And,

Theorem 2.1 The $E(Gut(G_{n})) (n\geq1)$ of the random polygonal chain $G_{n}$ is

Proof. The random polygonal chain $G_{n+1}$ is $G_{n}$ linked a new terminal polygonal $H_{n+1}$ by an edge, the $H_{n+1}$ is made up with vertices $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{2k}$, and the new edge is $u_{n}x_{1}$; see Figure 1. By(1.2), one has

Note that

Recall that $d(x_{1}) = 3$ and $d(x_{i}) = 2$ ($i\in\{2, 3, 4, \ldots, \ 2k\}$). On the basis of (2.1)-(2.3), We get

From (2.4), one has,

Then

For a random polygonal chain $G_{n}$, the number $\sum_{v\in V_{G_{n}}}d(v)d(u_{n}, v)$ is a random variable. We let

Substituting $A_{n}$ into (2.5), we obtain the recurrence formula of $E(Gut(G_{n}))$

We continue to consider the following $k$ possible ways.

Way 1. $G_{n}\longrightarrow G_{n+1}^{1}$. In this way, $u_{n}$ same as the vertex $x_{2}$ or $x_{2k}$. Then, $\sum_{v\in V_{G_{n}}}d(v)d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(v)d(x_{2}, v)$ or $\sum_{v\in V_{G_{n}}}d(v)d(x_{2k}, v)$ with probability $p_{1}$.

Way 2. $G_{n}\longrightarrow G_{n+1}^{2}$. In this way, $u_{n}$ same as the vertex $x_{3}$ or $x_{2k-1}$. Then, $\sum_{v\in V_{G_{n}}}d(v)d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(v)d(x_{3}, v)$ or $\sum_{v\in V_{G_{n}}}d(v)d(x_{2k-1}, v)$ with probability $p_{2}$.

Way 3. $G_{n}\longrightarrow G_{n+1}^{3}$. In this way, $u_{n}$ same as the vertex $x_{4}$ or $x_{2k-2}$. Then, $\sum_{v\in V_{G_{n}}}d(v)d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(v)d(x_{4}, v)$ or $\sum_{v\in V_{G_{n}}}d(v)d(x_{2k-2}, v)$ with probability $p_{3}$.

$\vdots$      $\vdots$          $\vdots$

Way k-3. $G_{n}\longrightarrow G_{n+1}^{k-3}$. In this way, $u_{n}$ same as the vertex $x_{k-2}$ or $x_{k+4}$. Then, $\sum_{v\in V_{G_{n}}}d(v)$ $d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(v)d(x_{k-2}, v)$ or $\sum_{v\in V_{G_{n}}}d(v)d(x_{k+4}, v)$ with probability $p_{k-3}$.

Way k-2. $G_{n}\longrightarrow G_{n+1}^{k-2}$. In this way, $u_{n}$ same as the vertex $x_{k-1}$ or $x_{k+3}$. Then, $\sum_{v\in V_{G_{n}}}d(v)$ $d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(v)d(x_{k-1}, v)$ or $\sum_{v\in V_{G_{n}}}d(v)d(x_{k+3}, v)$ with probability $p_{k-2}$.

Way k-1. $G_{n}\longrightarrow G_{n+1}^{k-1}$. In this way, $u_{n}$ same as the vertex $x_{k}$ or $x_{k+2}$. Then, $\sum_{v\in V_{G_{n}}}d(v)d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(v)d(x_{k}, v)$ or $\sum_{v\in V_{G_{n}}}d(v)d(x_{k+2}, v)$ with probability $p_{k-1}$.

Way k. $G_{n}\longrightarrow G_{n+1}^{k}$, then $u_{n}$ is the vertex $x_{k+1}$. Then, $\sum_{v\in V_{G_{n}}}d(v)d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(v)d(x_{k+1}, v)$ with probability $1-p_{1}-p_{2}-p_{3}-\cdots-p_{k-3}-p_{k-2}-p_{k-1}$.

On the basis of the above $k$ ways, we get

Substitute the expectation for the above equation, let $E(A_{n}) = A_{n}$, we obtain

Let

Hence,

By the calculation

Based on the above results, we have

Thus,

Substitute $A_{n}$ into (2.6), we have,

By the calculation $E(Gut(G_{1})) = 4k^{3}$.

Finally, we get the expected expression

as desired.

Specially, if $p_{1} = 1$, which implies $p_{2} = p_{3} = \cdots = p_{k} = 0$, then $G_{n}\cong M_{n}$. Similarly, if $p_{2} = 1$ (resp. $p_{3} = 1$, $\ldots$, $p_{k-2} = 1$, $p_{k-1} = 1$, ), which implies $p_{1} = p_{3} = \cdots = p_{k} = 0$ (resp. $p_{1} = p_{2} = p_{4} = \cdots = p_{k} = 0$, $p_{1} = \cdots = p_{k-3} = p_{k-1} = p_{k} = 0$, $p_{1} = p_{2}\cdots = p_{k-2} = p_{k} = 0$), then $G_{n}\cong O_{n}^{1}$ (resp. $G_{n}\cong O_{n}^{2}$, $\ldots$, $G_{n}\cong O_{n}^{k-2}$). If $p_{k} = 1$, which implies $p_{1} = p_{2} = \cdots = p_{k-1} = 0$, then $G_{n}\cong L_{n}$. According to Theorem 2.1 we can receive the Gutman index of the polygonal meta-chain $M_{n}$, the polygonal ortho-chain $O_{n}^{1}$, $O_{n}^{2}$, $O_{n}^{3}$, $\ldots$, $O_{n}^{k-2}$, the polygonal para-chain $L_{n}$, as

Obviously

Corollary 2.2 For a random polygonal chain $G_{n} \ (n\geq3)$, the para-chain $L_{n}$ gets to the maximum and the meta-chain $M_{n}$ gets to the minimum of $E(Gut(G_{n}))$.

Proof. By Theorem 2.1

When $n\geq3$, take the partial derivative of $E(Gut(G_{n})$

When $p_{1} = p_{2} = \cdots = p_{k-1} = 0$ (i.e. $p_{k} = 1$), the para-chain $L_{n}$ gets to the maximum of $E(Gut(G_{n}))$, that is $G_{n}\cong L_{n}$. If $p_{1}+p_{2}+p_{3}+\cdots+p_{k-1} = 1$, let $p_{k-1} = 1-p_{1}-p_{2} -\cdots-p_{k-2}\ (0\leq p_{1}\leq1, \ 0\leq p_{2}\leq1, \ldots, 0\leq p_{k-1}\leq1)$, we have

Therefore,

So $p_{1} = p_{2} = \cdots = p_{k-2} = 0$ (i.e. $p_{k-1} = 1$), $E(Gut(G_{n}))$ can't attain the minimum value ^[28]. With the same calculations as the same above, If $p_{1}+p_{2}+p_{3}+\ldots+p_{i} = 1$, let $p_{i} = 1-p_{1}-p_{2} -\ldots-p_{i-1}\ $$(0\leq p_{1}\leq1,$$\ 0\leq p_{2}\leq1, \ldots,$$0\leq p_{i-1}\leq1)$, $(i\geq3)$, we have

Therefore,

only when $p_{1}+p_{2} = 1$, they may get to the minimum value ^[29,30]. Then let $p_{1} = 1-p_{2} \ (0\leq p_{2}\leq1)$

Thus,

So $E(G(Gut_{n}))$ achieves the minimum value, when $p_{2} = 0 \ (i.e. \ p_{1} = 1)$, that is $G_{n}\cong M_{n}$.

3.   The Schultz index in a random polygonal chain

In this section, we will consider the expected values of Schultz index of the random polygonal chain. In fact, $G_{n+1}$ is $G_{n}$ linked to a new terminal polygonal $H_{n+1}$ by an edge, the $H_{n+1}$ is made up with vertices $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{2k}$, and the new edge is $u_{n}x_{1}$; see Figure 1.

Theorem 3.1 The $E(S(G_{n}))(n\geq1)$ of the random polygonal chain $G_{n}$ is

Proof. Recall the random polygonal chain $G_{n+1}$ is $G_{n}$ linked a new terminal polygonal $H_{n+1}$ by an edge, the $H_{n+1}$ is made up with vertices $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{2k}$, and the new edge is $u_{n}x_{1}$; see Figure 1. By(1.3),

Note that

Recall that $d(x_{1}) = 3$ and $d(x_{i}) = 2$ ($i\in\{2, 3, 4, \ldots, 2k\}$). On the basis of (2.1)-(2.3), We get

From (2.4), one has

Then

For a random polygonal chain $G_{n}$, the number $\sum_{v\in V_{G_{n}}}d(u_{n}, v)$ is a random variable. We let

Substituting $B_{n}$ into (3.1), we obtain the recurrence formula of $E(S(G_{n}))$,

Continue to consider the following $k$ possible ways.

Way 1. $G_{n}\longrightarrow G_{n+1}^{1}$. In this way, $u_{n}$ same as the vertex $x_{2}$ or $x_{2k}$. Then, $\sum_{v\in V_{G_{n}}}d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(x_{2}, v)$ or $\sum_{v\in V_{G_{n}}}d(x_{2k}, v)$ with probability $p_{1}$.

Way 2. $G_{n}\longrightarrow G_{n+1}^{2}$. In this way, $u_{n}$ same as the vertex $x_{3}$ or $x_{2k-1}$. Then, $\sum_{v\in V_{G_{n}}}d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(x_{3}, v)$ or $\sum_{v\in V_{G_{n}}}d(x_{2k-1}, v)$ with probability $p_{2}$.

Way 3. $G_{n}\longrightarrow G_{n+1}^{3}$. In this way, $u_{n}$ same as the vertex $x_{4}$ or $x_{2k-2}$. Then, $\sum_{v\in V_{G_{n}}}d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(x_{4}, v)$ or $\sum_{v\in V_{G_{n}}}d(x_{2k-2}, v)$ with probability $p_{3}$.

$\vdots$      $\vdots$          $\vdots$

Way k-3. $G_{n}\longrightarrow G_{n+1}^{k-3}$. In this way, $u_{n}$ same as the vertex $x_{k-2}$ or $x_{k+4}$. Then, $\sum_{v\in V_{G_{n}}}d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(x_{k-2}, v)$ or $\sum_{v\in V_{G_{n}}}d(x_{k+4}, v)$ with probability $p_{k-3}$.

Way k-2. $G_{n}\longrightarrow G_{n+1}^{k-2}$. In this way, $u_{n}$ same as the vertex $x_{k-1}$ or $x_{k+3}$. Then, $\sum_{v\in V_{G_{n}}}d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(x_{k-1}, v)$ or $\sum_{v\in V_{G_{n}}}d(x_{k+3}, v)$ with probability $p_{k-2}$.

Way k-1. $G_{n}\longrightarrow G_{n+1}^{k-1}$. In this way, $u_{n}$ same as the vertex $x_{k}$ or $x_{k+2}$. Then, $\sum_{v\in V_{G_{n}}}d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(x_{k}, v)$ or $\sum_{v\in V_{G_{n}}}d(x_{k+2}, v)$ with probability $p_{k-1}$.

Way k. $G_{n}\longrightarrow G_{n+1}^{k}$, then $u_{n}$ is the vertex $x_{k+1}$. Then, $\sum_{v\in V_{G_{n}}}d(u_{n}, v)$ is described as $\sum_{v\in V_{G_{n}}}d(x_{k+1}, v)$ with probability $1-p_{1}-p_{2}-p_{3}-\cdots-p_{k-3}-p_{k-2}-p_{k-1}$.

On the basis of the above $k$ ways, we get

Substitute the expectation for the above equation, , and let $E(B_{n}) = B_{n}$, we obtain

Let

Hence,

By the calculation

Based on the above results, we have

Thus,

and

Therefore,

and $E(S(G_{1})) = 4k^{3}$.

Finally, we get the expected expression

Let

Hence,

as desired.

Specially, If we set $(p_{1}, p_{2}, p_{3},$ $\ldots, p_{k-1}) = (1, 0, 0, \ldots, 0), (0, 1, 0, \ldots, 0),$ $(0, 0, 1, \ldots, 0),$ $\cdots, (0, \ldots, 1, 0, 0),$ $(0, \ldots, 0, 1, 0),$ $(0, \ldots, 0, 0, 1),$ $(0, \ldots, 0, 0, 0)$, respectively, and by Theorem 3.1, we can receive the Schultz index of the meta-chain ${M_{n}}$, the ortho-chain ${O_{n}^{1}}$, ${O_{n}^{2}}$, $\ldots$, ${O_{n}^{k-2}}$ and the para-chain ${L_{n}}$, as

Obviously

Corollary 3.2 For a random polygonal chain $G_{n} \ (n\geq3)$, the para-chain $L_{n}$ gets to the maximum and the meta-chain $M_{n}$ gets to the minimum of $E(S(G_{n}))$.

Proof. By Theorem 3.1

When $n\geq3$, take the partial derivative of $E(S(G_{n}))$

When $p_{1} = p_{2} = \ldots = p_{k-1} = 0$ (i.e. $p_{k} = 1$), the para-chain $L_{n}$ gets to the maximum of $E(S(G_{n}))$, that is $G_{n}\cong L_{n}$. If $p_{1}+p_{2}+p_{3}+\ldots+p_{k-1} = 1$, let $p_{k-1} = 1-p_{1}-p_{2} -\ldots-p_{k-2}\ (0\leq p_{1}\leq1, \ 0\leq p_{2}\leq1, \ldots, 0\leq p_{k-1}\leq1)$, we have

Therefore,

So $p_{1} = p_{2} = \ldots = p_{k-2} = 0$ (i.e. $p_{k-1} = 1$), $E(S(G_{n}))$can't attain the minimum value. With the same calculations as above, If $p_{1}+p_{2}$$+p_{3}+\ldots+p_{i} = 1$, let $p_{i} =$$1-p_{1}-p_{2} -\ldots-p_{i-1}\ (0\leq p_{1}\leq1,$$\ 0\leq p_{2}\leq1, \ldots,$$0\leq p_{i-1}\leq1)$, $(i\geq3)$, we have

Therefore,

only when $p_{1}+p_{2} = 1$, they may get to the minimum value. Then let $p_{1} = 1-p_{2} \ (0\leq p_{2}\leq1)$

Thus,

So $E(S(G_{n}))$ achieves the minimum value, when $p_{2} = 0 \ (i.e. \ p_{1} = 1)$, that is $G_{n}\cong M_{n}$.

4.   The average values for the the Gutman index and Schultz index

Recall that $\Theta_{n}$ is the set of all polygonal chains with $n$ polygons. We give the average value for the Gutman index and Schultz index with respect to $\Theta_{n}$.

For achieving the average value $Gut_{avr}(\Theta_{n})$, $S_{avr}(\Theta_{n})$, It takes $p_{1} = p_{2} = \ldots = p_{k} = \frac{1}{k}$ in the expected value for the Gutman index and Schultz index of the random polygonal chain (i.e.$E(Gut(G_{n}))$, $E(S(G_{n}))$). According to Theorem 2.1 and 3.1,

Theorem 4.1The $Gut_{avr}(\Theta_{n})(n\geq 1)$ and $S_{avr}(\Theta_{n}) (n\geq 1)$ for the the Gutman index and Schultz index of the random chain $G_{n}$ are

After verification, the equations are established,

5.   Conclusion

In this paper, we establish the explicit analytical expressions for the expected values of the Gutman index, Schultz index of a random polygonal chain. We also get the extremal values and average values of Gutman and Schultz indices. All these results will contribute to the study of the topological index of graphs and can better predict the physicochemical properties of more novel compounds, which can be applied to the research of drugs, macromolecular polymers and new materials ^{[20,31,32,33]}.

In chemical graph theory, the matter of polygonal chain is being widely studied by researchers. The molecular structures of polygonal chemicals are various and its physicochemical properties also become more and more important, and refer to ^{[34,35,36,37]}. It is possible to establish exact formulas for the expected values of some indices of a random polygon chain with $n$ regular polygons^{[38,39,40,41]}.

Acknowledgments

The authors wish to express their sincere appreciation to the editor and the anonymous referees for their valuable comments and suggestions. This research is supported by the National Science Foundation of China (Grant No.12171190), the Graduate innovation fund project of Anhui University of Science and Technology(Grant No.149), the Natural Science Foundation of Anhui Province(Grant No.2008085MA01) and Funded by Research Foundation of the Institute of Environment-friendly Materials and Occupational Health (Wuhu), Anhui University of Science and Technology((Grant No.ALW2021YF09)).

Conflict of interest

The authors declare that they have no competing interests.