
We construct a West Nile virus epidemic model that includes the interaction between the bird hosts and mosquito vectors, mosquito life stages (eggs, larvae, adults), and the dynamics of both larvicide and adulticide. We derive the basic reproduction number for the epidemic as the spectral radius of the next generation matrix. We formulate two impulsive optimal control problems which seek to balance the cost of insecticide applications (both the timing and application level) with the benefit of (1) vector control: reducing the number of mosquitoes or (2) disease control: reducing the disease burden. We reformulate these impulsive optimal control problems as nonlinear optimization problems and derive associated necessary conditions for the optimal controls. Numerical simulations are used to address three questions: How does the control and its impact on the system vary with the objective type? Is it beneficial to optimize the treatment timing? How does the control and its impact on the population vary with the type of pesticide used?
Citation: Folashade Agusto, Daniel Bond, Adira Cohen, Wandi Ding, Rachel Leander, Allis Royer. Optimal impulse control of West Nile virus[J]. AIMS Mathematics, 2022, 7(10): 19597-19628. doi: 10.3934/math.20221075
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We construct a West Nile virus epidemic model that includes the interaction between the bird hosts and mosquito vectors, mosquito life stages (eggs, larvae, adults), and the dynamics of both larvicide and adulticide. We derive the basic reproduction number for the epidemic as the spectral radius of the next generation matrix. We formulate two impulsive optimal control problems which seek to balance the cost of insecticide applications (both the timing and application level) with the benefit of (1) vector control: reducing the number of mosquitoes or (2) disease control: reducing the disease burden. We reformulate these impulsive optimal control problems as nonlinear optimization problems and derive associated necessary conditions for the optimal controls. Numerical simulations are used to address three questions: How does the control and its impact on the system vary with the objective type? Is it beneficial to optimize the treatment timing? How does the control and its impact on the population vary with the type of pesticide used?
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.
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