
In this paper, we introduced a family of distributions with a very flexible shape named generalized scale mixtures of generalized asymmetric normal distributions (GSMAGN). We investigated the main properties of the new family including moments, skewness, kurtosis coefficients and order statistics. A variant of the expectation maximization (EM)-type algorithm was established by combining the proflie likihood approach (PLA) with the classical expectation conditional maximization (ECM) algorithm for parameter estimation of this model. This approach with analytical expressions in the E-step and tractable M-step can greatly improve the computational speed and efficiency of the algorithm. The performance of the proposed algorithm was assessed by some simulation studies. The feasibility of the proposed methodology was illustrated through two real datasets.
Citation: Ruijie Guan, Aidi Liu, Weihu Cheng. The generalized scale mixtures of asymmetric generalized normal distributions with application to stock data[J]. AIMS Mathematics, 2024, 9(1): 1291-1322. doi: 10.3934/math.2024064
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In this paper, we introduced a family of distributions with a very flexible shape named generalized scale mixtures of generalized asymmetric normal distributions (GSMAGN). We investigated the main properties of the new family including moments, skewness, kurtosis coefficients and order statistics. A variant of the expectation maximization (EM)-type algorithm was established by combining the proflie likihood approach (PLA) with the classical expectation conditional maximization (ECM) algorithm for parameter estimation of this model. This approach with analytical expressions in the E-step and tractable M-step can greatly improve the computational speed and efficiency of the algorithm. The performance of the proposed algorithm was assessed by some simulation studies. The feasibility of the proposed methodology was illustrated through two real datasets.
In graph theory, graph labeling is an assignment of labels or weights to the vertices and edges of a graph. Graph labeling plays an important role in many fields such as computer science, coding theory and physics [32]. Baca et al. [10] have introduced the definition of an edge irregular total ℓ-labeling of any graph as a labeling L:V∪E→{1,2,3,…,ℓ} in which every two distinct edges fh and f∗h∗ of a graph G have distinct weights, this means that wL(fh)≠wL(f∗h∗) where wL(fh)=L(f)+L(h)+L(fh). They have deduced inequality which gives a lower bound of tes(G) for a graph G,
tes(G)≥max{⌈|E(G)|+23⌉,⌈Δ+12⌉} | (1) |
Also, they have introduced the exact value of TEIS, tes(G) for some families of graphs like fan graph Fn and wheel graph Wn,
tes(Fn)=⌈3n+23⌉, |
tes(Wn)=⌈2n+23⌉. |
In [15] authors have proved that for any tree T
tes(T)=max{⌈k+13⌉,⌈Δ+12⌉}, |
where Δ is maximum degree on k vertices. In addition, Salama [26] investigated the exact value of TEIS for a polar grid graph,
tes(Pm,n)=⌈2mn+23⌉. |
Authors in [1] determined TEIS for zigzag graphs. Also, the exact value of TEIS of the generalized web graph Wn,m and some families has been determined, see [14]. Tilukay et al. [31] have investigated total irregularity strength for a wheel graph, a fan graph, a triangular Book graph and a friendship graph. On the other hand, in [2,3,8,17,20,24,29] the total edge irregularity strengths for hexagonal grid graphs, centralized uniform theta graphs, generalized helm graph, series parallel graphs, disjoint union of isomorphic copies of generalized Petersen graph, disjoint union of wheel graphs, subdivision of star Sn and categorical product of two cycles have been investigated. For more details, see [4,5,6,7,9,11,12,13,16,18,19,21,23,25,27,28,30].
A generalized theta graph θ(t1,t2,…,tn) is a pair of n internal disjoint paths with lengths at least two joined by end vertices where the end vertices are named south pole S and north pole N and ti is the number of vertices in the nth path. Uniform theta graph θ(t,m) is a generalized theta graph in which all paths have the same numbers of internal vertices, for more details see [22].
In this paper, we have defined a new type of family of graph called uniform theta snake graph, θn(t,m). Also, the exact value of TEIS for some special types of the new family has been determined.
In the following, we define a new type of graph which is called uniform theta snake graph.
Definition 1. If we replace each edge of a path Pn by a uniform theta graph θ(t,m), we have a uniform theta snake graph θn(t,m). See Figure 1.
It is clear that for a uniform theta snake graph |E(θn(t,m))|=t(m+1)n and |V(θn(t,m))|=(tm+1)n+1. In this section, we determine the exact value of TEIS for uniform theta snake graph θn(3,3), θn(3,m), θn(t,3), θn(4,m), and θn(t,4).
Theorem 1. For a uniform theta snake graph θn(3,3) with 10n+1 vertices and 12n edges, we have
tes(θn(3,3))=4n+1. |
Proof. Since a uniform theta snake graph θn(3,3) has 12n edges and (θn(3,3))=6, then from (1) we have:
tes(θn(3,3))≥4n+1. |
To prove the invers inequality, we show that ħ− labeling is an edge irregular total for θn(3,3), see Figure 2, and ħ=4n+1. Let ħ=4n+1 and a total ħ− labeling α:V(θn(3,3))∪E(θn(3,3))→{1,2,3,…,ħ} is defined as:
α(c0)=1, |
α(cs)=4sfor1≤s≤n−1 |
α(cn)=ħ, |
α(xi,j)={jfor1≤j≤3j+1for4≤j≤6....j+n−1for3n−2≤j≤3n−1,i=1,2,3, |
α(xi,3n)=ħ−1fori=1,2,3 |
α(c0xi,1)=ifori=i1,2,3 |
α(cSxi,3S)=4S+ifor1≤S≤n−1,i=1,2,3 |
α(cSxi,3S+1)=4S+i+1for1≤S≤n−1,i=1,2,3 |
α(cnxi,3n)={ħ−2fori=1ħ−1fori=2ħfori=3, |
α(xi,jxi,j+1i)={j+i+1for1≤j≤2j+i+2for4≤j≤5....j+i+n−I1for3n−5≤j≤3n−4ħ+i−3for3n−2≤j≤3n−1,i=1,2,3. |
It is clear that ħ is the greatest used label. The weights of edges of θn(3,3) are given by:
wα(c0xi,1)=i+2fori=1,2,3, |
wα(cSxi,3S)=12S+i−1for1≤S≤n−1,i=1,2,3 |
wα(cSxi,3S+1)=12S+i+2for1≤S≤n−1,i=1,2,3, |
wα(cnxi,3n)={3(ħ−1)fori=13ħ−2fori=23ħ−1fori=3, |
wα(xi,jxi,j+1)={3j+i+2for1≤j≤23j+i+5for4≤j≤5....3j+i+3n−4for3n−5≤j≤3n−43ħ+i−10forj=3n−23ħ+i−7forj=3n−1,i=1,2,3 |
Obviously, the weights of edges are distinct. So α is an edge irregular total ħ− labeling. Hence
tes(θn(3,3))=4n+1. |
Theorem 2. For θn(3,m),m>3 be a uniform theta snake graph. Then
tes(θn(3,m))=(m+1)n+1. |
Proof. Since |E(θn(3,m))|=3(m+1)n and Δ(θn(3,m))=6. Substituting in (1), we find
tes(θn(3,m))≥(m+1)n+1. |
The existence of an edge irregular total ƛ− labeling for θn(3,m), See Figure 3, m>3 will be shown, with ƛ=(m+1)n+1. Define a total ƛ− labeling β:V(θn(3,m))∪E(θn(3,m))→{1,2,3,…,ƛ} for θn(3,m) as:
β(c0)=1, |
β(cs)=(m+1)sfor1≤s≤n−1, |
β(cn)=ƛ |
β(xi,j)={jfor1≤j≤mj+1form+1≤j≤2m....j+n−1form(n−1)+1≤j≤mn−1, |
β(xi,mn)=ƛ−1fori=1,2,3 |
β(c0xi,1)=1fori=1,2,3 |
β(cSxi,mS)=(m+1)S+ifor1≤S≤n−1,i=1,2,3 |
β(cSxi,mS+1)=(m+1)S+i+1for1≤S≤n−1,i=1,2,3 |
β(cnxi,mn)={ƛ−2fori=1ƛ−1fori=2ƛfori=3, |
β(xi,jxi,j+1)={j+i+1for1≤j≤m−1j+i+2form+1≤j≤2m−1....j+i+nform(n−1)+1≤j≤mn−2j+i+n−1forj=mn−1. |
Clearly, ƛ is the most label of edges and vertices. The edges weights are given as follows:
wβ(c0xi,1)=i+2fori=1,2,3, |
wβ(cSxi,mS)=3(m+1)S+i−1for1≤S≤n−1,i=1,2,3 |
wβ(cSxi,mS+1)=3(m+1)S+i+2for1≤S≤n−1,i=1,2,3, |
wβ(cnxi,mn)={3ƛ−3fori=13ƛ−2fori=23ƛ−1fori=3, |
wβ(xi,jxi,j+1)={3j+i+2for1≤j≤m−13j+i+5form+1≤j≤2m−1....3jI+i+3n−1form(n−1)+1≤j≤mn−23ƛ+i−7forj=mn−1, |
It is obvious that the weights of edges are different, thus β is an edge irregular total ƛ− labeling of θn(3,m). Hence
tes(θn(3,m))=(m+1)n+1. |
Theorem 3. Let θn(t,3) be a theta snake graph for t>3. Then
tes(θn(t,3))=⌈4tn+23⌉. |
Proof. A size of the graph θn(t,3) equals 4tn and Δ(θn(t,3))=2t, then from (1) we have
tes(θn(t,3))≥⌈4tn+23⌉. |
We define an edge irregular total ħ− labeling for θn(t,3) to get upper bound. So, let ħ=⌈4tn+23⌉ and a total ħ− labeling γ:V(θn(t,3))∪E(θn(t,3))→{1,2,3,…,ħ} is defined in the following three cases:
Case 1. 4tn+2≡0(mod3)
γ is defined as:
γ(c0)=1, |
γ(cS)=(t+1)Sfor1≤S≤n−1, |
γ(cn)=ħ |
γ(xi,j)I={ifor1≤j≤3,i=1,2,…,ti+t+1for4≤j≤6,i=1,2,…,ti+2(t+1)for7≤j≤9,i=1,2,…,t......i+(n−1)(t+1)for3n−5≤j≤3n−3,i=1,2,…,tħ−1for3n−2≤j≤3n,i=1ħfor3n−2≤j≤3n,i=2,3,…,t, |
γ(c0xi,1)=1fori=1,2,…,t |
γ(cSxi,3S)=2St−2S+3for1≤S≤n−1,i=1,2,…,t |
γ(cnxi,3n)={ħ−t+2fori=1ħ−t+ifori=2,3,…,t, |
γ(cSxi,3S+1)=2St−2S+2for1≤S≤n−1,i=1,2,…,t |
γ(cn−1xi,3n−2)={{(t+2)n−t−5fori=1(t+2)n−t+i−7fori=2,3,…,t,n=2,3{(t+1)n−t−1fori=1(t+1)n−t+i−3fori=2,3,…,t,n≠2,3 |
γ(xi,jxi,j+1)={{t+jfor1≤j≤23t+j−5for4≤j≤55t+j−10for7≤j≤8......(2n−3)t+j−5(n−2)for3n−5≤j≤3n−4,i=1,2,…,tħ−3(t+n)+j+5for3n−2≤j≤3n−1,i=1ħ−3(t+n)+j+5+2(i−2)for3n−2≤j≤3n−1,i=2,3,…,t |
Obviously, ħ is the greatest label. The edges weights of θn(t,3) can be expressed as:
wγ(c0xi,1)=i+2fori=1,2,…,t |
wγ(cSxi,3S)=t(4S−1)+i+2for1≤S≤n−1,i=1,2,…,t |
wγ(cSxi,3S+1)=4St+i+2for1≤S≤n−1,i=1,2,…,t, |
wγ(cn−1xi,3n−2)={2nt+3n−2t+ħ+i−8forn=2,32nt+2n−2t+ħ+i−4forn≠2,3,i=1,2,…,t |
wγ(xi,jxi,j+1)={{t+j+2ifor1≤j≤25t+j+2i−4for4≤j≤59t+j+2i−6for7≤j≤8......(4n−5)t+j+2i−3n+8for3ni−5≤j≤3n−4,i=1,2,…,t3ħ−3(t+in)+j+3for3n−2≤j≤3n−1,i=13ħ−3(t+in)+j+2i+3for3n−2≤j≤3n−1,i=2,3,…,t |
It implies that the edges weights have distinct values. So γ is the desired edge irregular total ħ− labeling, ħ=⌈4tn+23⌉. Hence
tes(θn(t,3))=⌈4tn+23⌉. |
Case 2. 4tn+2≡1(mod3)
Defineγ as:
γ(c0)=1, |
γ(cS)=(t+1)Sfor1≤S≤n−1, |
γ(cn)=ħ |
γ(xi,j)={ifor1≤j≤3,i=1,2,…,ti+t+1for4≤j≤6,i=1,2,…,ti+2(t+1)for7≤j≤9,i=1,2,…,t......i+(n+1)(t+1)for3n−5≤j≤3n−3,i=1,2,…,tħ−1for3n−2≤j≤3n,i=1ħfor3n−2≤j≤3n,i=2,3,…,t, |
γ(c0xi,1)=1fori=1,2,…,t |
γ(cSxi,3S)=2St−2S+3for1≤S≤n−1,i=1,2,…,t |
γ(cnxi,3n)={ħ−tfori=1ħ−t+i−2fori=2,3,…,t, |
γ(cSxi,3S+1)=2St−2S+2for1≤IS≤n−1,i=1,2,…t |
γ(cn−1xi3n−2)={{(t+2)n−t−5fori=1(t+2)n−t+i−7fori=2,3,…,t,n=2,3{(t+1)n−t−1fori=1(It+I1)n−t+i−3fori=2,3,…,t,n≠2,3 |
γ(xi,jxi,j+1)={{t+jfor1≤j≤23t+j−5for4≤j≤55t+j−10for7≤j≤8......(2n−3)t+j−5(n−2)for3n−5≤j≤3n−4,i=1,2,…,tħ−3(t+n)+j+3for3n−2≤j≤3n−1,i=1ħ−3(t+n)+j+2(i−2)for3n−2≤j≤3n−1,i=2,3,…,t |
It is clear that the greatest label is ħ. We define the weights of edges of θn(t,3) as:
wγ(c0xi,1)=i+2fori=1,2,…,t |
wγ(cSxi,3S)=t(4S−1)+i+2for1≤S≤n−1,i=1,2,…,twγ(cnxi,3n)=3ħ−t+i−2for1≤S≤n−1,i=1,2,…,t |
wγ(cSxi,3S+1)=4St+i+2for1≤S≤n−1,i=1,2,…,t, |
wγ(cn−1xi,3n−2)={2nt+3n−2t+ħ+i−8forn=2,32nt+2n−2t+ħ+i−4forn≠2,3,i=1,2,…,t |
wγ(xi,jxi,j+1)={{t+j+2ifor1≤j≤25t+j+2i−4for4≤j≤59t+j+2i−6for7≤j≤8......(4n−5)t+j+2i−3n+8for3n−5≤j≤3n−4,i=1,2,…,t3ħ−3(t+n)+j+1for3n−2≤j≤3n−1,i=13ħ−3(t+n)+j+2(i−2)for3n−2≤j≤3n−1,i=2,3,…,t |
It is obvious that the edges weights are different. Then
tes(θn(t,3))=⌈4tn+23⌉. |
Case 3. 4tn+2≡2(mod3)
γ is defined as follows:
γ(c0)=1, |
γ(cS)=(t+1)Sfor1≤S≤n−1, |
γ(cn)=ħ |
γ(xi,j)={ifor1≤j≤3,i=1,2,…,ti+t+1for4≤j≤6,i=1,2,…,ti+2(t+1)for7≤j≤9,i=1,2,…,t......i+(n−1)(t+1)for3n−5≤j≤3n−3,i=1,2,…,tħ−1for3n−2≤j≤3n,i=1ħfor3n−2≤j≤3n,i=2,3,…,t, |
γ(c0xi,1)=1fori=1,2,…,t |
γ(cSxi,3S)=2St−2S+3for1≤S≤n−1,i=1,2,…,t |
γ(cnxi,3n)={ħ−t+1fori=1ħ−t+i−1fori=2,3,…,t, |
γ(cSxi,3S+1)=2St−2S+2for1≤S≤n−2,i=1,2,…,t |
γ(cn−1xi,3n−2)={{(t+2)n−t−5fori=1(t+2)n−t+i−7fori=2,3,…,t,n=2,3{(t+1)n−t−1fori=1(t+1)n−t+i−3fori=2,3,…,t,n≠2,3 |
γ(xi,jxi,j+1)={{t+jfor1≤j≤23t+j−5for4≤j≤55t+j−10for7≤j≤8......(2n−3)t+j−5(n−2)for3n−5≤j≤3n−4,i=1,2,…tħ−3(t+i)+j+4for3n−2≤j≤3ni−1,i=1ħ−3(t+n)+j+2ifor3n−2≤j≤3n−1,i=2,3,…,t |
We can see that ħ is the greatest label. For edges weights of θn(t,3), we have
wγ(c0xi,1)=i+2fori=1,2,…,t |
wγ(c0xi,3S)=t(4S−1)+i+2for1≤S≤n−1,i=1,2,…,twγ(cnxi,3n)=3ħ−t+i−1for1≤S≤n−1,i=1,2,…,t |
wγ(cSxi,3S+1)=4St+i+2for1≤S≤n−1,i=1,2,…,t, |
wγ(cnxi,3n−2)={2nt−3n−2t+ħ+i−8forn=2,32nt+2n−2t+ħ+i−4forn≠2,3,i=1,2,…,t |
wγ(xi,jxi,j+1)={{t+j+2ifor1≤j≤25t+j+2i−4for4≤j≤59t+j+2i−6for7≤j≤8......(4n−5)t+j+2i−3n+8for3n−5≤j≤3n−4,i=1,2,…,t3ħ−3(t+n)+j+2for3n−2≤j≤3n−1,i=13ħ−3(t+n)+j+2ifor3n−2≤j≤3n−1,i=2,3,…,t |
It clears that the edges weights are i distinct. So γ is the desired edge irregular total ħ− labeling, ħ=⌈4tn+23⌉. Hence
tes(θn(t,3))=⌈4tn+23⌉. |
Theorem 4. For θn(4,m) be a theta snake graph for t>3. Then
tes(θn(4,m))=⌈4(m+1)n+23⌉. |
Proof. Since |E(θn(4,m))|=4(m+1)n and Δ(θn(4,m))=8, then from (1) we have
tes(θn(4,m))≥⌈4(m+1)n+23⌉. |
The existence of an edge irregular total ƛ− labeling for θn(4,m), m>3 will be shown, with ƛ=⌈4(m+1)n+23⌉. Define a total ƛ− labeling β:V(θn(4,m))∪E(θn(4,m))→{1,2,3,…,ƛ} for θn(4,m) in the following three cases as:
Case 1. 4(m+1)n+2≡0(mod3), i=1,2,3,4
β is defined as:
β(cs)={1fors=0(m+1)sfor1≤s≤⌈n2⌉ƛ+s−nfor⌈n2⌉≤s≤n, |
β(xi,j)={jfor1≤j≤mj+1form+1≤j≤2m....j+⌈n2⌉−1ƛ−j+22ƛform(⌈n2⌉−1)+1≤j≤m⌈n2⌉+1form⌈n2⌉+2≤j≤m(n−1)form(n−1)+1≤j≤mn−1, |
β(c0xi,1)=1fori=1,2,3,4 |
β(cSxi,mS)={2cS+i−1for1≤S≤⌈n2⌉−1cS+i−4(m+1)for⌈n2⌉≤s≤n−1ƛ−4+ifors=n,i=1,2,3,4 |
β(cSxi,mS+1)={2cS+i+1for1≤S≤⌈n2⌉,i=1,2,3,4cS+i−4(m+1)+2for⌈n2⌉+1≤s≤n−1 |
β(cnxi,mn)={ƛ−3fori=1ƛ−2fori=2ƛ−1ƛfori=3fori=4, |
β(xi,jxi,j+1)={j+i+1for1≤j≤m−1j+i+2form+1≤j≤2m−1....j+i+⌈n2⌉forj=m(⌈n2⌉−1)+12j+i−2[nm(⌈n2⌉−1)+1]form(⌈n2⌉−1)+2≤j≤mn−1. |
It is clear that ƛ is the greatest used label. The weights of edges of θn(4,m) are given by:
wβ(c0xi,1)=i+2fori=1,2,3,4, |
wβ(cSxi,mS)={2ms+s+2cS+i−1for1≤S≤⌈n2⌉−1,cS+i+ƛ+(s−4)(m+1)−n+⌈n2⌉−1for⌈n2⌉≤s≤n−13ƛ−4+i+s−nfors=n,i=1,2,3,4 |
wβ(cSxi,mS+1)={(2m+1)s+2cS+i+1for1≤S≤⌈n2⌉,2ƛ+s−n+cS+i−4(m+1)+2for⌈n2⌉≤s≤n−1i=1,2,3,4, |
wβ(cnxi,mn)={3ƛ+s−n−3fori=13ƛ+s−n−2fori=23ƛ+s−n−13ƛ+s−nfori=3fori=4, |
wβ(xi,jxi,j+1)={3j+i+2for1≤j≤m−13j+i+4form+1≤j≤2m−1....3j+i+3⌈n2⌉−1forj=m(⌈n2⌉−1)+14j+2ƛ+45+i−2[nm(⌈n2⌉−1)+1]2j+2ƛ+i−2[nm(⌈n2⌉−1)+1]form⌈n2⌉+2≤j≤m(n−1)form(n−1)+1≤j≤mn−1, |
It is obvious that the weights of edges are different, thus β is an edge irregular total ƛ− labeling of θn(4,m). Hence
tes(θn(4,m))=⌈4(m+1)n+23⌉. |
Case 2. 4(m+1)n+2≡1(mod3), i=1,2,3,4
β is defined as:
β(cs)={1fors=0(m+1)sfor1≤s≤⌈n2⌉ƛ+s−nfor⌈n2⌉≤s≤n, |
β(xi,j)={jfor1≤j≤mj+1form+1≤j≤2m....j+⌈n2⌉−1ƛ−j+22ƛform(⌈n2⌉−1)+1≤j≤m⌈n2⌉form⌈n2⌉+1≤j≤m(n−1)form(n−1)+1≤j≤mn−1, |
β(c0xi,1)=1fori=1,2,3,4 |
β(cSxi,mS)={2cS+i−1for1≤S≤⌈n2⌉−1,ƛ−7+ifors=⌈n2⌉cS+i−4m−2for⌈n2⌉+1≤s≤n−1ƛ−6+ifors=n,i=1,2,3,4 |
β(cSxi,mS+1)={2cS+i+1for1≤S≤⌈n2⌉cS+i−4mfor⌈n2⌉≤s≤n−1,i=1,2,3,4 |
β(cnxi,mn)={ƛ−5fori=1ƛ−4fori=2ƛ−3ƛ−2fori=3fori=4, |
β(xi,jxi,j+1)={j+i+1for1≤j≤m−1j+i+2form+1≤j≤2m−1....j+i+⌈n2⌉forj=m(⌈n2⌉−1)+12j+i−2[nm(⌈n2⌉−1)+1]form(⌈n2⌉−1)+2≤j≤mn−1. |
It is clear that ƛ is the greatest used label. The weights of edges of θn(4,m) are given by:
wβ(c0xi,1)=i+2fori=1,2,3,4, |
wβ(cSxi,mS)={2ms+s+2cS+i−1for1≤S≤⌈n2⌉−1,2ƛ−m⌈n2⌉+(m+1)s+i+15fors=⌈n2⌉cS+i+ƛ+(s−4)(m+1)−n+⌈n2⌉−1for⌈n2⌉≤s≤n−13ƛ−4+i+s−nfors=n |
wβ(cSxi,mS+1)={(2m+1)s+2cS+i+1for1≤S≤⌈n2⌉,i=1,2,3,42ƛ+s−n+cS+i−4mfor⌈n2⌉≤s≤n−1i=1,2,3,4, |
wβ(cnxi,mn)={3ƛ+s−n−5fori=13ƛ+s−n−4fori=23ƛ+s−n−33ƛ+s−n−2fori=3fori=4, |
wβ(xi,jxi,j+1)={3j+i+2for1≤j≤m−13j+i+4form+1≤j≤2m−1....3j+i+3⌈n2⌉−1forj=m(⌈n2⌉−1)+14j+2ƛ+45+i−2[nm(⌈n2⌉−1)+1]2j+2ƛ+i−2[nm(⌈n2⌉−1)+1]form⌈n2⌉+2≤j≤m(n−1)form(n−1)+1≤j≤mn−1, |
It is obvious that the weights of edges are different, thus β is an edge irregular total ƛ− labeling of θn(4,m). Hence
tes(θn(4,m))=⌈4(m+1)n+23⌉. |
Case 3. 4(m+1)n+2≡2(mod3), i=1,2,3,4
β is defined as:
β(cs)={1fors=0(m+1)sfor1≤s≤⌈n2⌉ƛ+s−nfor⌈n2⌉≤s≤n, |
β(xi,j)={jfor1≤j≤mj+1form+1≤j≤2m....j+⌈n2⌉−1ƛ−j+22ƛform(⌈n2⌉−2)+1≤j≤m(⌈n2⌉−1)form(⌈n2⌉−1)+1≤j≤m(n−1)form(n−1)+1≤j≤mn−1, |
β(c0xi,1)=1fori=1,2,3,4 |
β(cSxi,mS)={2cS+i−1for1≤S≤⌈n2⌉−1,i=1,2,3,4ƛ−7+ifors=⌈n2⌉cS+i−4m−2for⌈n2⌉+1≤s≤n−1ƛ−5+ifors=n |
β(cSxi,mS+1)={2cS+i+1for1≤S≤⌈n2⌉−1,i=1,2,3,4cS+1+ifors=⌈n2⌉cS+i−4m+1for⌈n2⌉+1≤s≤n−1 |
β(cnxi,mn)={ƛ−4fori=1ƛ−3fori=2ƛ−2ƛ−1fori=3fori=4, |
β(xi,jxi,j+1)={j+i+1for1≤j≤m−1j+i+2form+1≤j≤2m−1....j+i+⌈n2⌉forj=m(⌈n2⌉−1)+12j+i−2[nm(⌈n2⌉−1)+1]+1form(⌈n2⌉−1)+2≤j≤mn−1. |
It is clear that ƛ is the greatest used label. The weights of edges of θn(4,m) are given by:
wβ(c0xi,1)=i+2fori=1,2,3,4, |
wβ(cSxi,mS)={2ms+s+2cS+i−1for1≤S≤⌈n2⌉−1,2ƛ−m⌈n2⌉+(m+1)s+i+15fors=⌈n2⌉cS+i+ƛ+(s−4)(m+1)−n+⌈n2⌉−1for⌈n2⌉≤s≤n−13ƛ−3+i+s−nfors=n |
wβ(cSxi,mS+1)={(2m+1)s+2cS+i+1for1≤S≤⌈n2⌉,i=1,2,3,42ƛ+s−n+cS+i−4m+1for⌈n2⌉≤s≤n−1, |
wβ(cnxi,mn)={3ƛ+s−n−3fori=13ƛ+s−n−2fori=23ƛ+s−n−13ƛ+s−nfori=3fori=4, |
wβ(xi,jxi,j+1)={3j+i+2for1≤j≤m−13j+i+4form+1≤j≤2m−1....3j+i+3⌈n2⌉−1forj=m(⌈n2⌉−1)+14j+2ƛ+45+i−2[nm(⌈n2⌉−1)+1]2j+2ƛ+i−2[nm(⌈n2⌉−1)+1]form⌈n2⌉+2≤j≤m(n−1)form(n−1)+1≤j≤mn−1, |
It is obvious that the weights of edges are different, thus β is an edge irregular total ƛ− labeling of θn(4,m). Hence
tes(θn(4,m))=⌈4(m+1)n+23⌉ |
Theorem 5. If θn(t,4) is theta snake graph for t>3. Then
tes(θn(t,4))=⌈5tn+23⌉. |
Proof. Since |E(θn(t,4))|=5tn and Δ(θn(t,4))=2t. Substituting in (1), we have
tes(θn(t,4))≥⌈5tn+23⌉. |
We define an edge irregular total ħ− labeling for θn(t,4) to get upper bound. Let ħ=⌈5tn+23⌉ and a total ħ− labeling γ:V(θn(t,4))∪E(θn(t,4))→{1,2,3,…,ħ} is defined in the following three cases:
Case 1. 5tn+2≡0(mod3)
Defineγ as:
γ(c0)=1, |
γ(cS)=(t+1)Sfor1≤S≤n−1, |
γ(cn)=ħ |
γ(xi,j)={ifor1≤j≤4,i=1,2,…,ti+t+1for5≤j≤8,i=1,2,…,ti+2(t+1)for9≤j≤12,i=1,2,…,t......i+(n−1)(t+1)for4n−7≤j≤4n−4,i=1,2,…,tħ−1for4n−3≤j≤4n,i=1ħfor4n−3≤j≤4n,i=2,3,…,t, |
γ(c0xi,1)=1fori=1,2,…,t |
γ(cSxi,4S)=3St−2S+3for1≤S≤n−1,i=1,2,…,t |
γ(cnxi,4n)={ħ−t+2fori=1ħ−t+ifori=2,3,…,t, |
γ(cSxi,4S+1)=3St−2S+2for1≤S≤n−1,i=1,2,…,t |
γ(cn−1xi,4n−3)={{(t+2)n−t−5fori=1(t+2)n−t+i−7fori=2,3,…,t,n=2,3(t+1)n−t+i−3fori=2,3,…,t,n≠2,3 |
γ(xi,jxi,j+1)={{t+jfor1≤j≤23t+j−5for4≤j≤55t+j−10for7≤j≤8......(2n−3)t+j−5(n−2)for4n−5≤j≤4n−4,i=1,2,…,tħ−3(t+n)+j+5for4n−2≤j≤4n,i=1ħ−3(t+n)+j+5+2(i−2)for4n−2≤j≤4n,i=2,3,…,t |
It is clear that, ħ is the greatest label. The edges weights of θn(t,4) can be expressed as:
wγ(c0xi,1)=i+2fori=1,2,…,t |
wγ(cSxi,4S)=t(5S−1)+i+2for1≤S≤n−1,i=1,2,…,twγ(cnxi,4n)=3ħ−t+ifori=1,2,…,t |
wγ(cSxi,4S+1)=5St+i+2for1≤S≤n−1,i=1,2,…,t, |
wγ(cn−1xi,4n−2)={2nt+3n−2t+ħ+i−8forn=2,32nt+2n−2t+ħ+i−6forn≠2,3,i=1,2,…,t |
wγ(xi,jxi,j+1)={{t+j+2ifor1≤j≤25t+j+2i−4for4≤j≤59t+j+2i−6for7≤j≤8......(4n−5)t+j+2i−3n+8for4n−5≤j≤4n−4,i=1,2,…,t3ħ−3(t+n)+j+3for4n−2≤j≤4n−1,i=13ħ−3(t+n)+j+2i+3for4n−2≤j≤4n−1,i=2,3,…,t |
It implies that the edges weights have distinct values. So γ is the desired edge irregular total ħ− labeling, ħ=⌈5tn+23⌉. Hence
tes(θn(t,4))=⌈5tn+23⌉. |
Case 2. 5tn+2≡1(mod3)
Defineγ as:
γ(c0)=1, |
γ(cS)=(t+1)Sfor1≤S≤n−1, |
γ(cn)=ħ |
γ(xi,j)={ifor1≤j≤4,i=1,2,…,ti+t+1for5≤j≤8,i=1,2,…,ti+2(t+1)for9≤j≤12,i=1,2,…,t......i+(n+1)(t+1)for4n−7≤j≤4n−4,i=1,2,…,tħ−1for4n−3≤j≤4n,i=1ħfor4n−3≤j≤4n,i=2,3,…,t, |
γ(c0xi,1)=1fori=1,2,…,t |
γ(cSxi,4S)=3St−2S+3for1≤S≤n−1,i=1,2,…,t |
γ(cnxi,4n)={ħ−tfori=1ħ−t+i−2fori=2,3,…,t, |
γ(cSxi,4S+1)=3St−2S+2 |
for1≤S≤n−1,i=1,2,…t |
γ(cn−1xi,4n−3)={{(t+2)n−t−5fori=1(t+2)n−t+i−7fori=2,3,…,t,n=2,3{(t+1)n−t−1fori=1(t+1)n−t+i−3fori=2,3,…,t,n≠2,3 |
γ(xi,jxi,j+1)={{t+jfor1≤j≤33t+j−5for5≤j≤75t+j−10for9≤j≤11......(2n−3)t+j−5(n−2)for4n−7≤j≤4n−5,i=1,2,…,tħ−4(t+n)+j+3for4n−3≤j≤4n−1,i=1ħ−4(t+n)+j+2(i−2)for4n−3≤j≤4n−1,i=2,3,…,t |
It is clear that the i greatest label is ħ. We define the weights of edges of θn(t,4) as:
wγ(c0xi,1)=i+2fori=1,2,…,t |
wγ(cSxi,4S)=t(5S−1)+i+2for1≤S≤n−1,i=1,2,…,twγ(cnxi,4n)=3ħ−t+i−2for1≤S≤n−1,i=1,2,…,t |
wγ(cSxi,4S+1)=5St+i+2for1≤S≤n−1,i=1,2,…,t, |
wγ(cn−1xi,4n−3)={3nt+3n−2t+ħ+i−8forn=2,33nt+2n−2t+ħ+i−6forn≠2,3,i=1,2,…,t |
wγ(xi,jxi,j+1)={{t+j+2ifor1≤j≤35t+j+2i−4fori5≤j≤79t+j+2i−6for9≤j≤11......(4n−5)t+j+2i−3n+8for4n−7≤j≤4n−5,i=1,2,…,t3ħ−4(t+n)+j+1for4n−3≤j≤4n−1,i=13ħ−4(t+n)+j+2(i−2)for4n−3≤j≤4n−1,i=2,3,…,t |
It is obvious that the edges weights are different. Then
tes(θn(t,4))=⌈5tn+23⌉. |
Case 3. 5tn+2≡2(mod3)
Defineγ as:
γ(c0)=1, |
γ(cS)=(t+1)Sfor1≤S≤n−1, |
γ(cn)=ħ |
γ(xi,j)={ifor1≤j≤4,i=1,2,…,ti+t+1for5≤j≤8,i=1,2,…,ti+2(t+1)for9≤j≤12,i=1,2,…,t......i+(in−1)(t+1)for4n−7≤j≤4n−4,i=1,2,…,tħ−1for4n−3≤j≤4n,i=1ħfor4n−3≤j≤4n,i=2,3,…,t, |
γ(c0xi,1)=1fori=1,2,…,t |
γ(cSxi,4S)=3St−2S+3for1≤S≤n−1,i=1,2,…,t |
γ(cnxi,4n)={ħ−t+1fori=1ħ−t+i−1fori=2,3,…,t, |
γ(cSxi,4S+1)=3St−2S+2for1≤S≤n−2,i=1,2,…,t |
γ(cn−1xi,4n−3)={{(t+2)n−t−5fori=1(t+2)n−t+i−7fori=2,3,…,t,n=2,3{(t+1)n−t−1fori=1(t+1)n−t+i−3fori=2,3,…,t,n≠2,3 |
γ(xi,jxi,j+1)={{t+jfor1≤j≤33t+j−5for5≤j≤75t+j−10for9≤j≤11......(2n−3)t+j−5(n−2)for4n−7≤j≤4n−5,i=1,2,…tħ−4(t+n)+j+4for4n−3≤j≤4n−1,i=1ħ−4(t+n)+j+2ifor4n−3≤j≤4n−1,i=2,3,…,t |
We can see that ħ is the greatest label. For edges weights of θn(t,4), we have:
wγ(c0xi,1)=i+2fori=1,2,…,t |
wγ(c0xi,4S)=t(5S−1)+i+2for1≤S≤n−1,i=1,2,…,twγ(cnxi,4n)=3ħ−t+i−1for1≤S≤in−1,i=1,2,…,t |
wγ(cSxi,4S+1)=5St+i+2for1≤S≤n−1,i=1,2,…,t, |
wγ(cnxi,4n−3)={2nt−3n−2t+ħ+i−8forn=2,32nt+2n−2t+ħ+i−6forn≠2,3,i=1,2,…,t |
wγ(xi,jxi,j+1)={{t+j+2ifor1≤j≤35t+j+2i−4for5≤j≤79t+j+2i−6for9≤j≤11......(4n−5)t+j+2i−3n+8for4n−7≤j≤4n−5,i=1,2,…,t3ħ−4(t+n)+j+2for4n−3≤j≤3n−1,i=13ħ−4(t+n)+j+2ifor4n−3≤j≤4n−1,i=2,3,…,t |
It is obvious that the edges weights are distinct. So γ is the desired edge irregular total ħ− labeling, ħ=⌈5tn+23⌉. Hence
tes(θn(t,4))=⌈5tn+23⌉. |
The previous results lead us to introduce the following conjecture for a general case of a uniform theta snake graph θn(t,m).
The previous results lead us to introduce the following conjecture for a general case of a uniform theta snake graph θn(t,m).
Conjecture. For uniform theta snake graph θn(t,m), n≥2,t≥3,andm≥3 we have
tes(θn(t,m))=⌈(m+1)tn+23⌉. |
In the current paper, we have defined a new type of a family of graph called uniform theta snake graph, θn(t,m). Also, the exact i value of TEISs for θn(3,3), θn(3,m) and θn(t,3) has been determined. Finally, we have generalized for t, m and found TEIS of a uniform theta snake graph θn(t,m) for m≥3, t≥3.
tes(θn(3,3))=4n+1. |
tes(θn(3,im))=(im+1)in+1. |
tes(θn(t,3))=⌈4tn+23⌉ |
tes(θn(t,m))=⌈(m+1)tn+23⌉. |
All authors declare no conflict of interest in this paper.
We are so grateful to the reviewer for his many valuable suggestions and comments that significantly improved the paper.
[1] |
D. F. Andrews, C. L. Mallows, Scale mixtures of normal distribution, J. R. Stat. Soc. Ser. B Methodol., 36 (1974), 99–102. https://doi.org/10.1111/j.2517-6161.1974.tb00989.x doi: 10.1111/j.2517-6161.1974.tb00989.x
![]() |
[2] |
M. West, On scale mixtures of normal distributions, Biometrika, 74 (1987), 646–648. https://doi.org/10.1093/biomet/74.3.646 doi: 10.1093/biomet/74.3.646
![]() |
[3] |
M. D. Branco, D. K. Dey, A general class of multivariate skew-elliptical distributions, J. Multivar. Anal., 79 (2001), 99–113. https://doi.org/10.1006/jmva.2000.1960 doi: 10.1006/jmva.2000.1960
![]() |
[4] |
C. S. Ferreira, H. Bolfarine, V. H. Lachos, Linear mixed models based on skew scale mixtures of normal distribution, Commun. Stat. Simul. Comput., 51 (2020), 7194–7214. https://doi.org/10.1080/03610918.2020.1827265 doi: 10.1080/03610918.2020.1827265
![]() |
[5] |
R. M. Basso, V. H. Lachos, C. R. B. Cabral, P. Ghosh, Robust mixture modeling based on scale mixtures of skew-normal distributions, Comput. Stat. Data Anal., 54 (2010), 2926–2941. https://doi.org/10.1016/j.csda.2009.09.031 doi: 10.1016/j.csda.2009.09.031
![]() |
[6] |
H. M. Kim, M. G. Genton, Characteristic functions of scale mixtures of multivariate skew-normal distributions, Comput. Stat. Data Anal., 102 (2011), 1105–1117. https://doi.org/10.1016/j.jmva.2011.03.004 doi: 10.1016/j.jmva.2011.03.004
![]() |
[7] |
T. I. Lin, J. C. Lee, W. J. Hsieh, Robust mixture modeling using the skew t distribution, Stat. Comput., 17 (2007), 81–92. https://doi.org/10.1007/s11222-006-9005-8 doi: 10.1007/s11222-006-9005-8
![]() |
[8] |
T. I. Lin, H. J. Ho, C. R. Lee, Flexible mixture modelling using the multivariate skew-t-normal distribution, Stat. Comput., 24 (2014), 531–546. https://doi.org/10.1007/s11222-013-9386-4 doi: 10.1007/s11222-013-9386-4
![]() |
[9] |
I. Lin, J. C. Lee, Y. Y. Shu, Finite mixture modelling using the skew normal distribution, Stat. Sin., 17 (2007), 909–927. https://doi.org/10.2307/24307705 doi: 10.2307/24307705
![]() |
[10] |
A. Mahdavi, V. Amirzadeh, A. Jamalizadeh, T. I. Lin, Maximum likelihood estimation for scale-shape mixtures of flexible generalized skew normal distributions via selection representation, Comput. Stat., 36 (2021), 2201–2230. https://doi.org/10.1007/s00180-021-01079-2 doi: 10.1007/s00180-021-01079-2
![]() |
[11] | A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Stat., 12 (1985), 171–178. |
[12] |
A. Azzalini, A. Dalla Valle, The multivariate skew-normal distribution, Biometrika, 83 (1996), 715–726. https://doi.org/10.1093/biomet/83.4.715 doi: 10.1093/biomet/83.4.715
![]() |
[13] |
A. Azzalini, A. Capitanio, Statistical applications of the multivariate skew normal distribution, J. R. Stat. Soc. Ser. B Methodol., 61 (1999), 579–602. https://doi.org/10.1111/1467-9868.00194 doi: 10.1111/1467-9868.00194
![]() |
[14] |
R. B. Arellano-Valle, A. Azzalini, On the unification of families of skew-normal distributions, Scand. J. Stat., 33 (2006), 561–574. https://doi.org/10.1111/j.1467-9469.2006.00503.x doi: 10.1111/j.1467-9469.2006.00503.x
![]() |
[15] |
R. B. Arellano-Valle, M. G. Genton, On fundamental skew distributions, J. Multivar. Anal., 96 (2005), 93–116. https://doi.org/10.1016/j.jmva.2004.10.002 doi: 10.1016/j.jmva.2004.10.002
![]() |
[16] |
A. Azzalini, The skew-normal distribution and related multivariate families, Scand. J. Stat., 32 (2005), 159–188. https://doi.org/10.1111/j.1467-9469.2005.00426.x doi: 10.1111/j.1467-9469.2005.00426.x
![]() |
[17] |
C. Fernandez, M. F. J. Steel, Reference priors for non-normal two-sample problems, Test, 7 (1988), 179–205. https://doi.org/10.1007/BF02565109 doi: 10.1007/BF02565109
![]() |
[18] |
D. M. Zhu, V. Zinde-Walsh, Properties and estimation of asymmetric exponential power distribution, J. Econom., 148 (2009), 86–99. https://doi.org/10.1016/j.jeconom.2008.09.038 doi: 10.1016/j.jeconom.2008.09.038
![]() |
[19] |
R. J. Guan, X. Zhao, C. H. Cheng, Y. H. Rong, A new generalized t distribution based on a distribution construction method, Mathematics, 9 (2021), 2413. https://doi.org/10.1016/10.3390/math9192413 doi: 10.1016/10.3390/math9192413
![]() |
[20] | H. Exton, Handbook of hypergeometric integrals: Theory, applications, tables, computer programs, New York: Halsted Press, 1978. |
[21] |
A. P. Dempster, N. M. Laird, D. B. Rubin, Maximum likelihood from incomplete data via the em algorithm, J. R. Stat. Soc. Ser. B Methodol., 39 (1977), 1–22. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x doi: 10.1111/j.2517-6161.1977.tb01600.x
![]() |
[22] | K. Lange, The EM algorithm, New York: Springer, 2013. |
[23] |
X. L. Meng, D. B. Rubin, Maximum likelihood estimation via the ecm algorithm: A general framework, Biometrika, 80 (1993), 267–278. https://doi.org/10.2307/2337198 doi: 10.2307/2337198
![]() |
[24] | P. Huber, Robust statistics, New York: Wiley, 1981. |
[25] |
L. L. Wen, Y. J. Qiu, M. H. Wang, J. L. Yin, P. Y. Chen, Numerical characteristics and parameter estimation of finite mixed generalized normal distribution, Commun. Stat. Simul. Comput., 51 (2022), 3596–3620. https://doi.org/10.1080/03610918.2020.1720733 doi: 10.1080/03610918.2020.1720733
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