# A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group

Authors: Hanan Alolaiyan; Abdul Razaq; Awais Yousaf; Rida Zahra
Journal: Journal of Mathematics
 (2021)

---

## Abstract

This work deals with the well-known group-theoretic graphs called coset graphs for the modular groupG and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Zp, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.


---

## Body

## 1. Introduction

The study of groups via their actions has been a central theme in group theory, since the completion of the classification of simple groups in the 1980s. This most commonly takes the form of actions on vector spaces and similar commutative objects or on more elementary combinatorial objects. It is not an exaggeration to say that the modular groupG (see [1–4]) is the single most important infinite discrete group, through its myriad connections with number theory, geometry, and topology. There is a long and venerable history of studying its actions, particularly on finite sets, which goes back to before the turn of the twentieth century. The modular group G has two generators f and g, where f: x ⟶ −1/x and g: x ⟶ x − 1/x are linear fractional transformations. The finite presentation of G is <f, g: f2 = g3 = 1>. It means that it is a free product of C2 and C3. The linear fractional transformation h: x ⟶ 1/x extends G to G′ because it inverts f and g; that is, h2 = (fh)2 = (gh)2 = 1. Thus, extended modular group G′ has three generators f, g, and h and its finite presentation is <f, g, h: f2 = g3 = h2 = (fh)2 = (gh)2 = 1>.Graph theory has applications in various branches of mathematics [5, 6]. Several topological and algebraic structures can be studied in a more effective way by using graphs. Graphical techniques are specifically utilized to investigate the finitely generated groups. The graphs prove to be an effective and simple method to solve many mathematical problems [7–9].The use of graphs to represent group actions has a venerable history. Cayley [8] published the first work on this topic. Mathematicians like Coxeter [9], Burnside [10], Stothers [11], Everitt [12], Conder [13], Whitehead [14], and others provided pioneering works on graphical representations of groups. The action of a modular group on certain objects can be represented by a certain type of graphs, called coset graphs. These were introduced by Higman in 1978. Later, in 1983, Mushtaq [15] laid their foundation. These graphs consist of triangles connected to each other. The edges of triangles are permuted anticlockwise to represent by g. Each vertex of a triangle is connected by f to another vertex of the triangle (which may be the same triangle).Moreover, the vertices of the coset graph that are fixed byf and g are represented by heavy dots. Since (gh)2 = 1 implies hgh−1 = g−1−1, h turns around the direction of the triangles like reflection. Thus, we do not introduce h-edges in coset graphs, so that they remain simple.The action ofG′ on finite field Zp is not possible because f maps 0 to ∞. Thus, we add ∞ to Zp in order to make the action possible.Example 1.
Let us consider the action of modular group onZ19∪∞. The permutation representations f, g, and h are(1)f:0,∞13,161,1811,122,910,173,67,84,1415,5,g:∞,1,010,18,27,9,315,6,416,14,517,11,13812,h:0,∞10,213,35,416,611,712,817,915,14118.
The corresponding coset graph is shown in Figure1.Figure 1
The coset graph ofZ19∪∞.Definition 1.
LetΓ and Γ′ be two coset graphs; then Γ′ is a homomorphic copy of Γ if(i)
order ofΓ′ is less than the order of Γ, that is, Γ′<Γ,(ii)
u is a vertex in Γ such that (u) x = u for some x∈G, and then there exists some vertex u′∈Γ′ such that u′x=u.
Letq′ be a real quadratic irrational number; then q′=α+m/β, where m is a square-free natural number and α,α2+m/β,β=1. In [16], Mushtaq studied the group action of G on real quadratic fields and showed that the corresponding coset graphs are infinite. Figure 2 shows a small patch of these graphs.
Due to the emergence of infinite graphs, the action ofG on q′=α+m/β through coset graphs is not easy to study. Therefore, the action of G on Zp∪∞ becomes important. The coset graphs for Zp∪∞ are homomorphic copies of the infinite graphs for q′=α+m/β, where m≡n2modp, for any natural number n. For example, the coset graph shown in Figure 1 is the homomorphic copy of the coset graph for α+17/β because 17≡62mod19.
For further details about coset graphs, we refer the readers to [17–22].
The main contributions of this paper are as follows:(1)
A thorough study on the formation of homomorphic copies of coset graphs is presented(2)
We have developed a formula to compute all homomorphic copies of the closed path of rank 4Figure 2
An overview of the small portion of infinite graphs.


## 2. Closed Paths in Coset Graph

Definition 2.
A closed path in a coset graph containing a vertex fixed byfgm1fg−1m2,…,fgmk−1fg−1mk∈PSL2,Z, where m1,m2,…,mk∈N is called a closed path of rank k. It is denoted by m1,m2,…,mk. In [23], it has been proved that the rank of closed paths is always even.Remark 1.
Letx,y∈G and ux=u. Then vertex uy is fixed by y−1xy.
Suppose thatu1 and u2 are any two vertices in a closed path C, such that u1x1=u1 and u2x2=u2. Let u1x3=u2; then x1−1x3 also maps u1 to u2. Clearly, x3 and x1−1x3 are the only possible paths to travel from u1 to u2. By contraction of vertices u1 and u2, we mean u1 and u2 merge to form a node u=u1=u2 such that u=u1=u2 is fixed by both x3 and x1−1x3. This can be done by making a closed path C′ containing u such that ux3=u and then by applying x1−1x3 on u such that x1−1x3 ends at u. Consequently, a graph γ is evolved, which is a homomorphic copy of C. Note that, in addition to u1 and u2, there are some other pairs of vertices in C, which also compose γ by contraction. In fact, during the formation of γ by contracting u1 and u2, some more pairs also get contracted. How many are they? The following theorems help to calculate this number.Theorem 1 (see [24]).
Let a homomorphic copyγ of C be formed by contracting its vertices u1 and u2. Then γ is obtainable also by contracting the pair u1x,u2x for some x∈G.Theorem 2 (see [24]).
The number of pairs to obtainγ is equal to the number of elements x∈G, such that u1x and u2x lie in C.Example 2.
Consider a closed path (see Figure3) containing a vertex v which is a fixed point of fg4fg−13. Thus, it is a rank two closed path, denoted by 4,3.
Figure4 represents the homomorphic copy of 4,3 generated through contraction of vertices u and v.Figure 3
The closed path (4, 3).Figure 4
A homomorphic copy of4,3.


## 3. Formation of Homomorphic Copies through Contraction of Vertices

The coset graphs are made up of closed paths. The vertices of infinite graphs are contracted in a specific manner to evolve finite coset graphs. Therefore, a question arises: how many distinct homomorphic copies can be created by contracting all pairs in a closed path? In this work, we have developed a technique to find all homomorphic copies of the closed pathsm1,m2,m3,m4, where m1>m2>m3 and m2=m4, in coset graphs. Diagrammatically m1,m2,m3,m4, where m1>m2>m3 and m2=m4, is shown in Figure 5.Figure 5
The closed pathm1,m2,m3,m4, where m1>m2>m3 and m2=m4.In the remaining part of the paper, we denote the closed pathm1,m2,m3,m4, where m1>m2>m3 and m2=m4, by ψ. Throughout this paper, the mirror image of any homomorphic copy γ is denoted by γ∗. If x=fgκ1fgκ2,…,fgκn∈PSL2,Z, where κi=1 or −1, then let x∗=fg−κ1fg−κ2,…,fg−κn. If x fixes any vertex u, then the vertex fixed by x∗ is u∗.Remark 2.
Since (gh)2 = 1 implies hgh−1 = g−1, h turns around the direction of the triangles like reflection. If γ is obtained by contracting vertices u1 and u2 of any closed path C, then the mirror image γ∗ of γ can be created by contracting u1∗ and u2∗. It should be noted that u1 and u2 do not need to lie in the same closed path C. From Figure 5, we have(2)si∗=s3m1−i−1,tj∗=v3m2−j−1,uk∗=u3m3−k−1,vl∗=t3m4−l−1,for i=1,2,3,…,3m1, j=1,2,3,…,3m2, k=1,2,3,…,3m3, and l=1,2,3,…,3m4. Thus, for each vertex u in ψ, there exists a vertex u∗ in ψ.Remark 3.
Some homomorphic copies have symmetry about the vertical axis; that is, they have the same orientations as those of their mirror images. In other words, they are mirror images of themselves. The homomorphic copyγ of any circuit C having a vertex u fixed by x1,x2 has a symmetry about vertical axis if and only if γ contains a vertex u∗ fixed by x1∗,x2∗.


### 3.1. Proposed Scheme

Sinceψ has 3m1+m2+m3+m4 number of vertices, the total number of pairs in ψ is 3m1+m2+m3+m42. We contract a pair of vertices u1,u2 of ψ such that a homomorphic copy γ is obtained. By using Theorem 2, we find all pairs of vertices in ψ, which form γ; let those be n in numbers. Now, we have two possibilities:(i)
If by contractingu1 and u2 vertices u1∗ and u2∗ are not contracted, then γ does not possess a vertical symmetry. Therefore, γ∗ consumes n more pairs of vertices of ψ.(ii)
Ifu1,u2 and u1∗,u2∗ are contracted all together, then γ has a symmetry about vertical axis. Therefore, γ∗ does not consume any pair. Thus, γ has n pairs of vertices.Next, we contractψ at one of the remaining pairs and the process continues until all 3m1+m2+m3+m42 pairs are exhausted.Leti1=0,1,2,…,m2−1. First, we contract vertex s3m1 with vertices s3i1+1 and the following result is obtained.Theorem 3.
If vertexs3m1 is contracted with vertices s3i1+1, then m2 distinct homomorphic copies αi1 of ψ are obtained. Furthermore, there are 3m22+3m2−2 pairs of vertices for these homomorphic copies.Proof.
LetD1=αi1:i1=0,1,2,…,m2−1 be the set of homomorphic copies of ψ obtained by contracting s3m1 with s3i1+1. In Figure 5, it can be seen that fgm2fg−1m3fgm4fg−1i1f and g−1fg−1m1−i1−1 are the possible paths between s3m1 and s3i1+1. Therefore, vertex s3m1=s3i1+1 in D1 is fixed by fgm2fg−1m3fgm4fg−1i1f and g−1fg−1m1−i1−1. It is also clear from Figure 5 that E1=f,fg−1,fg,fgf,fgfg−1,fg2,…,fgi1,fgi1f,fgi1fg−1,fgi1+1,g,g−1,e is the set of elements x of the modular group such that, for all x∈E1, both s3m1x and s3i1+1x lie in ψ. Since E1=3i1+2, by Theorem 2, there are 3i1+2 pairs in ψ to form αi1.
Next, we show that all homomorphic copiesαi1 of ψ in D1 are different and no copy of these is a mirror image of another.
Letαj,αk∈D1; then αj is evolved by contracting s3m1 and s3j+1, whereas αk is obtained by contracting s3m1 and s3k+1. Now αj=αk if and only if there exists an element x in E1 such that s3m1x=s3m1 and s3j+1x=s3k+1. One can see that only e∈E1 maps s3m1 to itself, but s3j+1e≠s3k+1.
Now suppose thatαj=αk∗; then there must exist some x in E1 which sends s3m1 to s3m1∗=s1 and s3j+1 to s3k+1∗=s3m1−k. But E1 does not contain such element. This means that all diagrams in D1 are distinct. Thus, D1=m2 and there are 3∑i1=0m2−1i1+2 pairs of vertices to create αi1.
Now we check how many diagrams inD1 have a symmetry about vertical axis. For this, let αj=αj∗; then E1 contains an element x such that(3)s3m1x=s3m1∗=s1,s3j+1x=s3j+1∗=s3m1−j,ors3m1x=s3j+1∗=s3m1−j,s3j+1x=s3m1∗=s1.
This is possible only ifi1=0; in this case, we have e∈E1 such that s3m1e=s1∗ and s1e=s3m1∗=s1. So, we conclude that only α0 has a symmetry about vertical axis; that is, α0 and its mirror image α0∗ have the same orientations, and all other m2−1 homomorphic copies in ψ do not possess a vertical symmetry. Hence there are 6∑i2=1m2−1i2+2+6=3m22+3m2−2 pairs to form D1=αi1:i1=0,1,2,…,m2−1.
We obtain all the results by using the same technique, so, from now onwards, instead of providing proofs of the theorems, we will present tables, which give the necessary information of the family of homomorphic copies evolved.
Leti2=1,2,…,m2−1. We contract the pair t3m2 and t3i2+1 and obtain the following result. By using Theorem 1, one can see that these vertices are not contracted in Theorem 3.Theorem 4.
If vertext3m2 is contracted with vertices t3i2+1, then m2−1 distinct homomorphic copies αi2′ of ψ are obtained. Furthermore, there are 3m22+3m2−4 pairs of vertices for these homomorphic copies.Table1 shows the complete information of the family of homomorphic copies αi2′. The information provided in Table 1 can be verified by the same technique used in the proof of Theorem 3.Let us contractu3m3 with u3i3+1, where i3=0,1,2,…,m3−1. By using Theorem 1, it can be easily verified that these pairs of vertices are not utilized in the previous theorems.Table 1
Necessary information regarding homomorphic copiesαi2′.
The pairs that are contractedt3m2,t3i2+1Family of homomorphic copies evolvedD2=αi2′D2m2−1Number of pairs for eachαi2′3i2+2Number of homomorphic copies that have vertical axis of symmetry0Total number of pairs forD23m22+3m2−4Theorem 5.
If vertexu3m3 is contracted with vertices u3i3+1, then m3 distinct homomorphic copies αi3″ of ψ are obtained. Furthermore, there are 3m32+3m3−2 pairs of vertices for these homomorphic copies.All details of the family of homomorphic copiesαi3″ have been provided in Table 2.Next, we contracts3m1 with t3i1+1 and formulate the following Theorem.Table 2
Necessary information regarding homomorphic copiesαi3″.
The pairs that are contractedu3m3,u3i3+1Family of homomorphic copies evolvedD3=αi3″D3m3Number of pairs for eachαi3″3i3+2Homomorphic copies possessing vertical axis of symmetryOnlyα0″Total number of pairsD33m32+3m3−2Theorem 6.
If vertexs3m1 is contracted with vertices t3i1+1, then m2 distinct homomorphic copies βi1 of ψ are obtained. Furthermore, there are 3m22+3m2 pairs of vertices for these homomorphic copies.For complete information ofβi1, see Table 3.Now we contract vertexs3m1 with vertices u3i3+1, where i3=0,1,2,…,m3−1, and acquire the following result.Table 3
Necessary information regarding homomorphic copiesβi1.
The pairs that are contracteds3m1,t3i1+1Family of homomorphic copies evolvedD4=βi1D4m2Number of pairs for eachβi1m2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD43m22+3m2Theorem 7.
If vertexs3m1 is contracted with vertices u3i3+1, then m3 distinct homomorphic copies βi3′ of ψ are obtained. Furthermore, there are 3m32+3m3 pairs of vertices for these homomorphic copies.Table4 provides all information regarding homomorphic copies βi3′, evolved in Theorem 7.The following theorem is evolved by contractings3m1 with v3i4+1, where i4=0,1,2,…,m4−1.Table 4
Necessary information regarding homomorphic copiesβi3′.
The pairs that are contracteds3m1,u3i3+1Family of homomorphic copies evolvedD5=βi3′D5m3Number of pairs for eachβi3′3i3+2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD53m32+3m3Theorem 8.
If vertexs3m1 is contracted with vertices v3i4+1, then m4 distinct homomorphic copies βi4″ of ψ are obtained. Furthermore, there are 3/2m42+3m4 pairs of vertices for these homomorphic copies.The complete information of the family of homomorphic copiesβi4″ obtained in Theorem 8 is given in Table 5.The next theorem is obtained by contractingt3m2 with s3i5+1, where i5=1,2,…,m1−1.Table 5
Necessary information regarding homomorphic copiesβi4″.
The pairs that are contracteds3m1,v3i4+1Family of homomorphic copies evolvedD6=βi4″D6m4Number of pairs for eachβi4″3i4+2Homomorphic copies possessing vertical axis of symmetryAllβi4″Total number of pairs forD63/2m42+3m4Theorem 9.
If vertext3m2 is contracted with vertices s3i5+1, then m1−1 distinct homomorphic copies βi5″′ of ψ are obtained. Furthermore, there are 3/2m12+3m1−4 pairs of vertices for these homomorphic copies.All details of the generated homomorphic copies are given in Table6.Now we contract vertext3m2 with vertices u3i6+1 for all i6=1,2,…,m3−1.Table 6
Necessary information regarding homomorphic copiesβi5″.
The pairs that are contractedt3m2,s3i5+1Family of homomorphic copies evolvedD7=βi5″D7m1−1Number of pairs for eachβi5″3i5+2Homomorphic copies possessing vertical axis of symmetryAllβi5″′Total number of pairs forD73/2m12+3m1−4Theorem 10.
If vertext3m2 is contracted with vertices u3i6+1, then m3−1 distinct homomorphic copies βi6iv of ψ are obtained. Furthermore, there are 3m32+3m3−4 pairs of vertices for these homomorphic copies.The complete information of the family of homomorphic copiesβi6iv is provided in Table 7.In the following theorem,u3m3 is contracted with t3i1+1.Table 7
Necessary information regarding homomorphic copiesβi6iv.
The pairs that are contractedt3m2,u3i6+1Family of homomorphic copies evolvedD8=βi6ivD8m3−1Number of pairs for eachβi6iv3i6+2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD83m32+3m3−4Theorem 11.
If vertexu3m3 is contracted with vertices t3i1+1, then m2 distinct homomorphic copies βi1v of ψ are obtained. Furthermore, there are 3/2m22+3m2 pairs of vertices for these homomorphic copies.All details of homomorphic copiesβi1v can be found in Table 8.Let us now contract vertexv3m4 with vertices u3i6+1, where i6=1,2,…,m3−1.Table 8
Necessary information regarding homomorphic copiesβi1v.
The pairs that are contractedu3m3,t3i1+1Family of homomorphic copies evolvedD9=βi1vD9m2Number of pairs for eachβi1v3i1+2Homomorphic copies possessing vertical axis of symmetryAllβi1vTotal number of pairs forD93/2m22+3m2Theorem 12.
If vertexv3m4 is contracted with vertices u3i6+1, then m3−1 distinct homomorphic copies βi6vi of ψ are obtained. Furthermore, there are 3/2m32+3m3−4 pairs of vertices for these homomorphic copies.All information related to the homomorphic copies created in Theorem12 is given in Table 9.Let us contract vertexs3m1 with vertices s3i7+1, where i7=m2+1,m2+2,…,m1−1. Consequently, we have the following theorem.Table 9
Necessary information regarding homomorphic copiesβi6vi.
The pairs that are contractedv3m4,u3i6+1Family of homomorphic copies evolvedD10=βi6viD10m3−1Number of pairs for eachβi6vi3i6+2Homomorphic copies possessing vertical axis of symmetryAllβi6viTotal number of pairs forD103/2m32+3m3−4Theorem 13.
If vertexs3m1 is contracted with vertices s3i7+1, then m1−m2−1 distinct homomorphic copies γi7 of ψ are obtained. Furthermore, there are 6m2+2m1−m2−1 pairs of vertices for these homomorphic copies.Table10 completely describes the family of homomorphic copies evolved in Theorem 11.Suppose thatε1=0,ifm1+m3is even1,ifm1+m3is odd and i8=m3+1,m3+2,…,m1+m3−ε1/2. Let us contract v3m4 with s3i8+1 to obtain Theorem 14.Table 10
Necessary information regarding homomorphic copiesγi7.
The pairs that are contracteds3m1,s3i7+1Family of homomorphic copies evolvedD11=γi7D11m1−m2−1Number of pairs for eachγi73m2+2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD116m2+2m1−m2−1Theorem 14.
If vertexv3m4 is contracted with vertices s3i8+1, then 1/2m1−m3−ε1 distinct homomorphic copies γi8′ of ψ are obtained. Furthermore, there are 3m3+2m1−m3−1 pairs of vertices for these homomorphic copies.All information of the homomorphic copies evolved in Theorem14 is provided in Table 11.The following theorem emerges as a result of contractingv3m4 with t3i9+1, where i9=m3+1,m3+2,…,m2−1.Table 11
Necessary information regarding homomorphic copiesγi8′.
The pairs that are contractedv3m4,s3i8+1Family of homomorphic copies evolvedD12=γi8′D121/2m1−m3−ε1Number of pairs for eachγi8′3m3+2Homomorphic copies possessing vertical axis of symmetryNo one ifm1+m3 is odd and γm1+m3/2′ if m1+m3 is evenTotal number of pairs forD123m3+2m1−m3−1Theorem 15.
If vertexv3m4 is contracted with vertices t3i9+1, then m2−m3−1 distinct homomorphic copies γi9″ of ψ are obtained. Furthermore, there are 6m3+2m2−m3−1 pairs of vertices for these homomorphic copies.Table12 shows the complete information of all homomorphic copies created in Theorem 15.Suppose thati10=1,2,…,m1−1 and i11=1,2,…,m2−1. Let us contract s3i10+1 with t3i11and obtain the following results.Table 12
Necessary information regarding homomorphic copiesγi9″.
The pairs that are contractedv3m4,t3i9+1Family of homomorphic copies evolvedD13=γi9″D13m2−m3−1Number of pairs for eachγi9″3m3+2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD136m3+2m2−m3−1Theorem 16.
If verticess3i10+1 are contracted with vertices t3i11, then m1−1m2−1 distinct homomorphic copies δi10,i11 of ψ are obtained. Furthermore, there are 12m1−1m2−1 pairs of vertices for these homomorphic copies.All necessary details of the family of homomorphic copies evolved in this process are provided in Table13.Leti12=1,2,3,…,m3−ε2−2/2, where ε2=0,ifm3is even1,ifm3is odd. The following results have been constructed by contracting s3i10+1 with u3i12.Table 13
Necessary information regarding homomorphic copiesδi10,i11.
The pairs that are contracteds3i10+1,t3i11Family of homomorphic copies evolvedD14=δi10,i11D14m1−1m2−1Number of pairs for eachδi10,i116Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD1412m1−1m2−1Theorem 17.
If verticess3i10+1 are contracted with vertices u3i12, then 1/2m1−1m3−ε2−2 distinct homomorphic copies δi10,i12′ of ψ are obtained. Furthermore, there are 6m1−1m3−ε2−2 pairs of vertices for these homomorphic copies.See Table14 for complete information of the homomorphic copies.Recall thati13=1,2,…,m1−1 and ε2=0ifm3is even1ifm3is odd. Let ε3=0ifm1is even1ifm1is odd, and we obtain the following theorem by contracting s3i13+1 with u3m3−ε2/2.Table 14
Necessary information regarding homomorphic copiesδi10,i12′.
The pairs that are contracteds3i10+1,u3i12Family of homomorphic copies evolvedD15=δi10,i12′D151/2m1−1m3−ε2−2Number of pairs for eachδi10,i12′6Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD156m1−1m3−ε2−2Theorem 18.
If verticess3i13+1 are contracted with vertex u3m3−ε2/2, then m1−ε2/2ifm3is evenm1−1ifm3is odd distinct homomorphic copies δi13,m3−i3/2″ of ψ are obtained. Furthermore, there are 6m1−1ifm3is even12m1−1ifm3is odd pairs of vertices for these homomorphic copies.All information related to the homomorphic copies created in Theorem18 is provided in Table 15.Now we contractt3i14+1 with u3i15, where i14=1,2,…,m2−1 and i15=1,2,…,m3−1, so that Theorem 19 is evolved.Table 15
Necessary information regarding homomorphic copiesδi13,m3−i3/2″.
The pairs that are contracteds3i13+1,u3m3−ε2/2Family of homomorphic copies evolvedD16=δi13,m3−i3/2″D16m1−ε2/2ifm3is evenm1−1ifm3is oddNumber of pairs for eachδi13,m3−i3/2″6Homomorphic copies possessing vertical axis of symmetryδm1/2,m3/2″ if m1 and m3 are even and no one otherwiseTotal number of pairs forD166m1−1ifm3is even12m1−1ifm3is oddTheorem 19.
If verticest3i14+1 are contracted with vertices u3i15, then m2−1m3−1 distinct homomorphic copies δi14,i15″′ of ψ are obtained. Furthermore, there are 12m2−1m3−1 pairs of vertices for these homomorphic copies.For complete details about the homomorphic copies evolved in Theorem19, see Table 16.Leti16=1,2,3,…,m4−i14. The following theorem is obtained by contracting t3i14+1 with v3i16.Table 16
Necessary information regarding homomorphic copiesδi14,i15″.
The pairs that are contractedt3i14+1,u3i15Family of homomorphic copies evolvedD17=δi14,i15″D17m2−1m3−1Number of pairs for eachδi14,i15″6Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD1712m2−1m3−1Theorem 20.
If verticest3i14+1 are contracted with vertices v3i16, then 1/2m2m2−1 distinct homomorphic copies δi14,i16iv of ψ are obtained. Furthermore, there are 6m2−12 pairs of vertices for these homomorphic copies.All important information regardingδi14,i16iv is provided in Table 17.Recall thatε3=0ifm1is even1ifm1is odd, and let i17=1,2,3,…,m1−2−ε3/2 and i18=i17+1,i17+2,…,m1−i17. In Theorem 21, we contract vertices s3i17+1 with vertices s3i18 to create homomorphic copies μi17,i18.Table 17
Necessary information regarding homomorphic copiesδi14,i16iv.
The pairs that are contractedt3i14+1,v3i16Family of homomorphic copies evolvedD18=δi14,i16ivD181/2m2m2−1Number of pairs for eachδi14,i16iv6Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD186m2−12Theorem 21.
If verticess3i17+1 are contracted with vertices s3i18, then 1/4m12−2m1+ε3 distinct homomorphic copies μi17,i18 of ψ are obtained. Furthermore, there are 3m12−3m1+2 pairs of vertices for these homomorphic copies.All information about the family of homomorphic copies evolved in Theorem21 is given in Table 18.Now, let us contract verticest3i19+1 with t3i20, where i19=1,2,3,…,m2−2 and i20=i19+1,i19+2,i19+3,…,m2−1.Table 18
Necessary information regarding homomorphic copiesμi17,i18.
The pairs that are contracteds3i17+1,s3i18Family of homomorphic copies evolvedD19=μi17,i18D191/4m12−2m1+ε3Number of pairs for eachμi17,i186Homomorphic copies possessing vertical axis of symmetry1/2m1−2+ε3Total number of pairs forD193m12−3m1+2Theorem 22.
If verticest3i19+1 are contracted with vertices t3i20, then 1/2m2−2m2−1 distinct homomorphic copies μi19,i20′ of ψ are obtained. Furthermore, there are 6m2−2m2−1 pairs of vertices for these homomorphic copies.Table19 provides all information regarding μi19,i20′.Recall thatε2=0ifm3is even1ifm3is odd and let i21=1,2,3,…,m3−2+ε2/2 and i22=i21+1,i21+2,i21+3,…,m3−i21. The next theorem is obtained by contracting u3i21+1 with u3i22Table 19
Necessary information regarding homomorphic copiesμi19,i20′.
The pairs that are contractedt3i19+1,t3i20Family of homomorphic copies evolvedD20=μi19,i20′D201/2m2−2m2−1Number of pairs for eachμi19,i20′6Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD206m2−2m2−1Theorem 23.
If verticesu3i21+1 are contracted with vertices u3i22, then 1/4m32−2m3+ε2 distinct homomorphic copies μi21,i22″ of ψ are obtained. Furthermore, there are 3m32−3m3+2 pairs of vertices for these homomorphic copies.All details of the generated homomorphic copies are given in Table20.Now suppose thati23=1,2,3,…,m1−ε3/2. The following results have been constructed by contracting s3i23+1 with s3i23.Table 20
Necessary information regarding homomorphic copiesμi21,i22″.
The pairs that are contractedu3i21+1,u3i22:i21=1,2,3,…,m3−2+ε2/2 and i22=i21+1,i21+2,i21+3,…,m3−i21Family of homomorphic copies evolvedD21=μi21,i22″D211/4m32−2m3+ε2Number of pairs for eachμi21,i22″6Homomorphic copies possessing vertical axis of symmetry1/2m3−2+ε2Total number of pairs forD213m32−3m3+2Theorem 24.
If verticess3i23+1 are contracted with vertices s3i23, then 1/2m1−ε3 distinct homomorphic copies νi23,i23 of ψ are obtained. Furthermore, there are 3m1−1 pairs of vertices for these homomorphic copies.See Table21 for complete details of the evolved homomorphic copies.Now, we contractt3i24+1 with t3i24, where i24=1,2,3,…,m2−1.Table 21
Necessary information regarding homomorphic copiesνi23,i23.
The pairs that are contracteds3i23+1,s3i23Family of homomorphic copies evolvedD22=νi23,i23D221/2m1−ε3Number of pairs for eachνi23,i233Homomorphic copies possessing vertical axis of symmetryνm1/2,m1/2Total number of pairs forD223m1−1Theorem 25.
If verticest3i24+1 are contracted with vertices t3i24, then m2−1 distinct homomorphic copies νi24,i24′ of ψ are obtained. Furthermore, there are 6m2−1 pairs of vertices for these homomorphic copies.The complete information of the family of homomorphic copiesνi24,i24′ obtained in Theorem 25 is given in Table 22.Suppose thati25=1,2,3,…,m3−ε2/2. Let us contract u3i25+1 and u3i25 and obtain the following results.Table 22
Necessary information regarding homomorphic copiesνi24,i24′.
The pairs that are contractedt3i24+1,t3i24Family of homomorphic copies evolvedD23=νi24,i24′D23m2−1Number of pairs for eachνi24,i24′3Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD236m2−1Theorem 26.
If verticesu3i25+1 are contracted with vertices u3i25, then 1/2m3−ε2 distinct homomorphic copies νi25,i25″ of ψ are evolved. Furthermore, there are 3m3−1 pairs of vertices for these homomorphic copies.All information of the family of homomorphic copiesνi25,i25″ evolved in Theorem 26 is provided in Table 23.Lastly, we contract 4 pairs of vertices to evolve 4 homomorphic copies.Table 23
Necessary information regarding homomorphic copiesνi25,i25″.
The pairs that are contractedu3i25+1,u3i25Family of homomorphic copies evolvedD24=νi25,i25″D241/2m3−ε2Number of pairs for eachνi25,i25″3Homomorphic copies possessing vertical axis of symmetryνm3/2,m3/2″Total number of pairs forD243m3−1Theorem 27.
(i)
Letη1 be the homomorphic copy of ψ obtained by contracting s1 and t1. Then there are 6m2+m3+2 pairs of vertices for η1.(ii)
Letη2 be the homomorphic copy obtained by contracting s1 and u1. Then there are 3m2+2m3+2 pairs of vertices for η2.(iii))
Letη3 be the homomorphic copy obtained by contracting s1 and v1. Then there are 6m2+1 pairs of vertices to generate η3.(iv)
Letη4 be the homomorphic copy obtained by contracting t1 and u1.Then there are 6m3+1 pairs of vertices for η4.
LetΓ=0ifm1andm3are odd4ifm1andm3are even−1ifm1is even andm3is odd or vice versa. We are now in a position to prove the main theorem of this paper.Theorem 28.
There are1/4m12+4m22+m32+4m1m2+m2m3+2m1m2+8m2+4m3+Γ numbers of distinct homomorphic copies of ψ by contracting all the pairs in ψ.Proof.
Let us contract the following pairs of vertices:(i)
s3m1 and s3i1+1(ii)
t3m2 and t3i2+1(iii)
u3m3 and u3i3+1(iv)
s3m1 and t3i1+1(v)
s3m1 and u3i3+1(vi)
s3m1 and v3i4+1(vii)
t3m2 and s3i5+1(viii)
t3m2 and u3i6+1(ix)
u3m3 and t3i1+1(x)
v3m4 and u3i6+1(xi)
s3m1 and s3i7+1(xii)
v3m4 and s3i8+1(xiii)
v3m4 and t3i9+1(xiv)
s3i10+1 and t3i11(xv)
s3i10+1 and u3i12(xvi)
s3i13+1 and u3m3−ε2/2(xvii)
t3i14+1 and u3i15(xviii)
t3i14+1 and v3i16(xix)
s3i17+1 and s3i18(xx)
t3i19+1 and t3i20(xxi)
u3i21+1 and u3i22(xxii)
s3i23+1 and s3i23(xxiii)
t3i24+1 and t3i24(xxiv)
u3i25+1 and u3i25(xxv)
s1 and t1(xxvi)
s1 and u1(xxvii)
s1 and v1(xxviii)
t1 and u1
Then, by Theorems3 to 27, we obtain the set(4)F=αi1,αi2′,αi3″,βi1,βi3′,βi4″,βi5″′,βi6iv,βi1v,βi6vi,γi7,γi8′,γi9″,δi10,i11,δi10,i12′,δi13,m3−i3/2″,δi14,i15″,δi14,i16iv,μi17,i18,μi19,i20′,μi21,i22″,νi23,i23,νi24,i24′,νi25,i25″,η1,η2,η3,η4,of homomorphic copies of ψ and there are(5)S=3m22+3m2−2+3m22+3m2−4+3m32+3m3−2+3m22+3m2+3m32+3m3+32m42+3m4+32m12+3m1−4+3m32+3m3−4+32m22+3m2+32m32+3m3−4+6m2+2m1−m2−1+3m3+2m1−m3−1+6m3+2m2−m3−1+12m1−1m2−1+6m1−1m3−ε2−2+6m1−1ifm3is even12m1−1ifm3is odd+12m2−1m3−1+6m2−12+3m12+3m1−2+6m2−2m2−1+3m32−3m3−2+3m1−1+6m3−1+3m3−1+6m2+m3+2+3m2+2m3+2+6m2+1+6m3−1,pairs to form F. S=3m1+m2+m3+m42 is the total number of pairs in ψ and(6)F=14m12+4m22+m32+4m1m2+m2m3+2m1m2+8m2+4m3+Γ.
Thus, there are(7)14m12+4m22+m32+4m1m2+m2m3+2m1m2+8m2+4m3+Γ,numbers of distinct homomorphic copies obtained by contracting all pairs in ψ.


## 3.1. Proposed Scheme

Sinceψ has 3m1+m2+m3+m4 number of vertices, the total number of pairs in ψ is 3m1+m2+m3+m42. We contract a pair of vertices u1,u2 of ψ such that a homomorphic copy γ is obtained. By using Theorem 2, we find all pairs of vertices in ψ, which form γ; let those be n in numbers. Now, we have two possibilities:(i)
If by contractingu1 and u2 vertices u1∗ and u2∗ are not contracted, then γ does not possess a vertical symmetry. Therefore, γ∗ consumes n more pairs of vertices of ψ.(ii)
Ifu1,u2 and u1∗,u2∗ are contracted all together, then γ has a symmetry about vertical axis. Therefore, γ∗ does not consume any pair. Thus, γ has n pairs of vertices.Next, we contractψ at one of the remaining pairs and the process continues until all 3m1+m2+m3+m42 pairs are exhausted.Leti1=0,1,2,…,m2−1. First, we contract vertex s3m1 with vertices s3i1+1 and the following result is obtained.Theorem 3.
If vertexs3m1 is contracted with vertices s3i1+1, then m2 distinct homomorphic copies αi1 of ψ are obtained. Furthermore, there are 3m22+3m2−2 pairs of vertices for these homomorphic copies.Proof.
LetD1=αi1:i1=0,1,2,…,m2−1 be the set of homomorphic copies of ψ obtained by contracting s3m1 with s3i1+1. In Figure 5, it can be seen that fgm2fg−1m3fgm4fg−1i1f and g−1fg−1m1−i1−1 are the possible paths between s3m1 and s3i1+1. Therefore, vertex s3m1=s3i1+1 in D1 is fixed by fgm2fg−1m3fgm4fg−1i1f and g−1fg−1m1−i1−1. It is also clear from Figure 5 that E1=f,fg−1,fg,fgf,fgfg−1,fg2,…,fgi1,fgi1f,fgi1fg−1,fgi1+1,g,g−1,e is the set of elements x of the modular group such that, for all x∈E1, both s3m1x and s3i1+1x lie in ψ. Since E1=3i1+2, by Theorem 2, there are 3i1+2 pairs in ψ to form αi1.
Next, we show that all homomorphic copiesαi1 of ψ in D1 are different and no copy of these is a mirror image of another.
Letαj,αk∈D1; then αj is evolved by contracting s3m1 and s3j+1, whereas αk is obtained by contracting s3m1 and s3k+1. Now αj=αk if and only if there exists an element x in E1 such that s3m1x=s3m1 and s3j+1x=s3k+1. One can see that only e∈E1 maps s3m1 to itself, but s3j+1e≠s3k+1.
Now suppose thatαj=αk∗; then there must exist some x in E1 which sends s3m1 to s3m1∗=s1 and s3j+1 to s3k+1∗=s3m1−k. But E1 does not contain such element. This means that all diagrams in D1 are distinct. Thus, D1=m2 and there are 3∑i1=0m2−1i1+2 pairs of vertices to create αi1.
Now we check how many diagrams inD1 have a symmetry about vertical axis. For this, let αj=αj∗; then E1 contains an element x such that(3)s3m1x=s3m1∗=s1,s3j+1x=s3j+1∗=s3m1−j,ors3m1x=s3j+1∗=s3m1−j,s3j+1x=s3m1∗=s1.
This is possible only ifi1=0; in this case, we have e∈E1 such that s3m1e=s1∗ and s1e=s3m1∗=s1. So, we conclude that only α0 has a symmetry about vertical axis; that is, α0 and its mirror image α0∗ have the same orientations, and all other m2−1 homomorphic copies in ψ do not possess a vertical symmetry. Hence there are 6∑i2=1m2−1i2+2+6=3m22+3m2−2 pairs to form D1=αi1:i1=0,1,2,…,m2−1.
We obtain all the results by using the same technique, so, from now onwards, instead of providing proofs of the theorems, we will present tables, which give the necessary information of the family of homomorphic copies evolved.
Leti2=1,2,…,m2−1. We contract the pair t3m2 and t3i2+1 and obtain the following result. By using Theorem 1, one can see that these vertices are not contracted in Theorem 3.Theorem 4.
If vertext3m2 is contracted with vertices t3i2+1, then m2−1 distinct homomorphic copies αi2′ of ψ are obtained. Furthermore, there are 3m22+3m2−4 pairs of vertices for these homomorphic copies.Table1 shows the complete information of the family of homomorphic copies αi2′. The information provided in Table 1 can be verified by the same technique used in the proof of Theorem 3.Let us contractu3m3 with u3i3+1, where i3=0,1,2,…,m3−1. By using Theorem 1, it can be easily verified that these pairs of vertices are not utilized in the previous theorems.Table 1
Necessary information regarding homomorphic copiesαi2′.
The pairs that are contractedt3m2,t3i2+1Family of homomorphic copies evolvedD2=αi2′D2m2−1Number of pairs for eachαi2′3i2+2Number of homomorphic copies that have vertical axis of symmetry0Total number of pairs forD23m22+3m2−4Theorem 5.
If vertexu3m3 is contracted with vertices u3i3+1, then m3 distinct homomorphic copies αi3″ of ψ are obtained. Furthermore, there are 3m32+3m3−2 pairs of vertices for these homomorphic copies.All details of the family of homomorphic copiesαi3″ have been provided in Table 2.Next, we contracts3m1 with t3i1+1 and formulate the following Theorem.Table 2
Necessary information regarding homomorphic copiesαi3″.
The pairs that are contractedu3m3,u3i3+1Family of homomorphic copies evolvedD3=αi3″D3m3Number of pairs for eachαi3″3i3+2Homomorphic copies possessing vertical axis of symmetryOnlyα0″Total number of pairsD33m32+3m3−2Theorem 6.
If vertexs3m1 is contracted with vertices t3i1+1, then m2 distinct homomorphic copies βi1 of ψ are obtained. Furthermore, there are 3m22+3m2 pairs of vertices for these homomorphic copies.For complete information ofβi1, see Table 3.Now we contract vertexs3m1 with vertices u3i3+1, where i3=0,1,2,…,m3−1, and acquire the following result.Table 3
Necessary information regarding homomorphic copiesβi1.
The pairs that are contracteds3m1,t3i1+1Family of homomorphic copies evolvedD4=βi1D4m2Number of pairs for eachβi1m2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD43m22+3m2Theorem 7.
If vertexs3m1 is contracted with vertices u3i3+1, then m3 distinct homomorphic copies βi3′ of ψ are obtained. Furthermore, there are 3m32+3m3 pairs of vertices for these homomorphic copies.Table4 provides all information regarding homomorphic copies βi3′, evolved in Theorem 7.The following theorem is evolved by contractings3m1 with v3i4+1, where i4=0,1,2,…,m4−1.Table 4
Necessary information regarding homomorphic copiesβi3′.
The pairs that are contracteds3m1,u3i3+1Family of homomorphic copies evolvedD5=βi3′D5m3Number of pairs for eachβi3′3i3+2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD53m32+3m3Theorem 8.
If vertexs3m1 is contracted with vertices v3i4+1, then m4 distinct homomorphic copies βi4″ of ψ are obtained. Furthermore, there are 3/2m42+3m4 pairs of vertices for these homomorphic copies.The complete information of the family of homomorphic copiesβi4″ obtained in Theorem 8 is given in Table 5.The next theorem is obtained by contractingt3m2 with s3i5+1, where i5=1,2,…,m1−1.Table 5
Necessary information regarding homomorphic copiesβi4″.
The pairs that are contracteds3m1,v3i4+1Family of homomorphic copies evolvedD6=βi4″D6m4Number of pairs for eachβi4″3i4+2Homomorphic copies possessing vertical axis of symmetryAllβi4″Total number of pairs forD63/2m42+3m4Theorem 9.
If vertext3m2 is contracted with vertices s3i5+1, then m1−1 distinct homomorphic copies βi5″′ of ψ are obtained. Furthermore, there are 3/2m12+3m1−4 pairs of vertices for these homomorphic copies.All details of the generated homomorphic copies are given in Table6.Now we contract vertext3m2 with vertices u3i6+1 for all i6=1,2,…,m3−1.Table 6
Necessary information regarding homomorphic copiesβi5″.
The pairs that are contractedt3m2,s3i5+1Family of homomorphic copies evolvedD7=βi5″D7m1−1Number of pairs for eachβi5″3i5+2Homomorphic copies possessing vertical axis of symmetryAllβi5″′Total number of pairs forD73/2m12+3m1−4Theorem 10.
If vertext3m2 is contracted with vertices u3i6+1, then m3−1 distinct homomorphic copies βi6iv of ψ are obtained. Furthermore, there are 3m32+3m3−4 pairs of vertices for these homomorphic copies.The complete information of the family of homomorphic copiesβi6iv is provided in Table 7.In the following theorem,u3m3 is contracted with t3i1+1.Table 7
Necessary information regarding homomorphic copiesβi6iv.
The pairs that are contractedt3m2,u3i6+1Family of homomorphic copies evolvedD8=βi6ivD8m3−1Number of pairs for eachβi6iv3i6+2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD83m32+3m3−4Theorem 11.
If vertexu3m3 is contracted with vertices t3i1+1, then m2 distinct homomorphic copies βi1v of ψ are obtained. Furthermore, there are 3/2m22+3m2 pairs of vertices for these homomorphic copies.All details of homomorphic copiesβi1v can be found in Table 8.Let us now contract vertexv3m4 with vertices u3i6+1, where i6=1,2,…,m3−1.Table 8
Necessary information regarding homomorphic copiesβi1v.
The pairs that are contractedu3m3,t3i1+1Family of homomorphic copies evolvedD9=βi1vD9m2Number of pairs for eachβi1v3i1+2Homomorphic copies possessing vertical axis of symmetryAllβi1vTotal number of pairs forD93/2m22+3m2Theorem 12.
If vertexv3m4 is contracted with vertices u3i6+1, then m3−1 distinct homomorphic copies βi6vi of ψ are obtained. Furthermore, there are 3/2m32+3m3−4 pairs of vertices for these homomorphic copies.All information related to the homomorphic copies created in Theorem12 is given in Table 9.Let us contract vertexs3m1 with vertices s3i7+1, where i7=m2+1,m2+2,…,m1−1. Consequently, we have the following theorem.Table 9
Necessary information regarding homomorphic copiesβi6vi.
The pairs that are contractedv3m4,u3i6+1Family of homomorphic copies evolvedD10=βi6viD10m3−1Number of pairs for eachβi6vi3i6+2Homomorphic copies possessing vertical axis of symmetryAllβi6viTotal number of pairs forD103/2m32+3m3−4Theorem 13.
If vertexs3m1 is contracted with vertices s3i7+1, then m1−m2−1 distinct homomorphic copies γi7 of ψ are obtained. Furthermore, there are 6m2+2m1−m2−1 pairs of vertices for these homomorphic copies.Table10 completely describes the family of homomorphic copies evolved in Theorem 11.Suppose thatε1=0,ifm1+m3is even1,ifm1+m3is odd and i8=m3+1,m3+2,…,m1+m3−ε1/2. Let us contract v3m4 with s3i8+1 to obtain Theorem 14.Table 10
Necessary information regarding homomorphic copiesγi7.
The pairs that are contracteds3m1,s3i7+1Family of homomorphic copies evolvedD11=γi7D11m1−m2−1Number of pairs for eachγi73m2+2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD116m2+2m1−m2−1Theorem 14.
If vertexv3m4 is contracted with vertices s3i8+1, then 1/2m1−m3−ε1 distinct homomorphic copies γi8′ of ψ are obtained. Furthermore, there are 3m3+2m1−m3−1 pairs of vertices for these homomorphic copies.All information of the homomorphic copies evolved in Theorem14 is provided in Table 11.The following theorem emerges as a result of contractingv3m4 with t3i9+1, where i9=m3+1,m3+2,…,m2−1.Table 11
Necessary information regarding homomorphic copiesγi8′.
The pairs that are contractedv3m4,s3i8+1Family of homomorphic copies evolvedD12=γi8′D121/2m1−m3−ε1Number of pairs for eachγi8′3m3+2Homomorphic copies possessing vertical axis of symmetryNo one ifm1+m3 is odd and γm1+m3/2′ if m1+m3 is evenTotal number of pairs forD123m3+2m1−m3−1Theorem 15.
If vertexv3m4 is contracted with vertices t3i9+1, then m2−m3−1 distinct homomorphic copies γi9″ of ψ are obtained. Furthermore, there are 6m3+2m2−m3−1 pairs of vertices for these homomorphic copies.Table12 shows the complete information of all homomorphic copies created in Theorem 15.Suppose thati10=1,2,…,m1−1 and i11=1,2,…,m2−1. Let us contract s3i10+1 with t3i11and obtain the following results.Table 12
Necessary information regarding homomorphic copiesγi9″.
The pairs that are contractedv3m4,t3i9+1Family of homomorphic copies evolvedD13=γi9″D13m2−m3−1Number of pairs for eachγi9″3m3+2Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD136m3+2m2−m3−1Theorem 16.
If verticess3i10+1 are contracted with vertices t3i11, then m1−1m2−1 distinct homomorphic copies δi10,i11 of ψ are obtained. Furthermore, there are 12m1−1m2−1 pairs of vertices for these homomorphic copies.All necessary details of the family of homomorphic copies evolved in this process are provided in Table13.Leti12=1,2,3,…,m3−ε2−2/2, where ε2=0,ifm3is even1,ifm3is odd. The following results have been constructed by contracting s3i10+1 with u3i12.Table 13
Necessary information regarding homomorphic copiesδi10,i11.
The pairs that are contracteds3i10+1,t3i11Family of homomorphic copies evolvedD14=δi10,i11D14m1−1m2−1Number of pairs for eachδi10,i116Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD1412m1−1m2−1Theorem 17.
If verticess3i10+1 are contracted with vertices u3i12, then 1/2m1−1m3−ε2−2 distinct homomorphic copies δi10,i12′ of ψ are obtained. Furthermore, there are 6m1−1m3−ε2−2 pairs of vertices for these homomorphic copies.See Table14 for complete information of the homomorphic copies.Recall thati13=1,2,…,m1−1 and ε2=0ifm3is even1ifm3is odd. Let ε3=0ifm1is even1ifm1is odd, and we obtain the following theorem by contracting s3i13+1 with u3m3−ε2/2.Table 14
Necessary information regarding homomorphic copiesδi10,i12′.
The pairs that are contracteds3i10+1,u3i12Family of homomorphic copies evolvedD15=δi10,i12′D151/2m1−1m3−ε2−2Number of pairs for eachδi10,i12′6Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD156m1−1m3−ε2−2Theorem 18.
If verticess3i13+1 are contracted with vertex u3m3−ε2/2, then m1−ε2/2ifm3is evenm1−1ifm3is odd distinct homomorphic copies δi13,m3−i3/2″ of ψ are obtained. Furthermore, there are 6m1−1ifm3is even12m1−1ifm3is odd pairs of vertices for these homomorphic copies.All information related to the homomorphic copies created in Theorem18 is provided in Table 15.Now we contractt3i14+1 with u3i15, where i14=1,2,…,m2−1 and i15=1,2,…,m3−1, so that Theorem 19 is evolved.Table 15
Necessary information regarding homomorphic copiesδi13,m3−i3/2″.
The pairs that are contracteds3i13+1,u3m3−ε2/2Family of homomorphic copies evolvedD16=δi13,m3−i3/2″D16m1−ε2/2ifm3is evenm1−1ifm3is oddNumber of pairs for eachδi13,m3−i3/2″6Homomorphic copies possessing vertical axis of symmetryδm1/2,m3/2″ if m1 and m3 are even and no one otherwiseTotal number of pairs forD166m1−1ifm3is even12m1−1ifm3is oddTheorem 19.
If verticest3i14+1 are contracted with vertices u3i15, then m2−1m3−1 distinct homomorphic copies δi14,i15″′ of ψ are obtained. Furthermore, there are 12m2−1m3−1 pairs of vertices for these homomorphic copies.For complete details about the homomorphic copies evolved in Theorem19, see Table 16.Leti16=1,2,3,…,m4−i14. The following theorem is obtained by contracting t3i14+1 with v3i16.Table 16
Necessary information regarding homomorphic copiesδi14,i15″.
The pairs that are contractedt3i14+1,u3i15Family of homomorphic copies evolvedD17=δi14,i15″D17m2−1m3−1Number of pairs for eachδi14,i15″6Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD1712m2−1m3−1Theorem 20.
If verticest3i14+1 are contracted with vertices v3i16, then 1/2m2m2−1 distinct homomorphic copies δi14,i16iv of ψ are obtained. Furthermore, there are 6m2−12 pairs of vertices for these homomorphic copies.All important information regardingδi14,i16iv is provided in Table 17.Recall thatε3=0ifm1is even1ifm1is odd, and let i17=1,2,3,…,m1−2−ε3/2 and i18=i17+1,i17+2,…,m1−i17. In Theorem 21, we contract vertices s3i17+1 with vertices s3i18 to create homomorphic copies μi17,i18.Table 17
Necessary information regarding homomorphic copiesδi14,i16iv.
The pairs that are contractedt3i14+1,v3i16Family of homomorphic copies evolvedD18=δi14,i16ivD181/2m2m2−1Number of pairs for eachδi14,i16iv6Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD186m2−12Theorem 21.
If verticess3i17+1 are contracted with vertices s3i18, then 1/4m12−2m1+ε3 distinct homomorphic copies μi17,i18 of ψ are obtained. Furthermore, there are 3m12−3m1+2 pairs of vertices for these homomorphic copies.All information about the family of homomorphic copies evolved in Theorem21 is given in Table 18.Now, let us contract verticest3i19+1 with t3i20, where i19=1,2,3,…,m2−2 and i20=i19+1,i19+2,i19+3,…,m2−1.Table 18
Necessary information regarding homomorphic copiesμi17,i18.
The pairs that are contracteds3i17+1,s3i18Family of homomorphic copies evolvedD19=μi17,i18D191/4m12−2m1+ε3Number of pairs for eachμi17,i186Homomorphic copies possessing vertical axis of symmetry1/2m1−2+ε3Total number of pairs forD193m12−3m1+2Theorem 22.
If verticest3i19+1 are contracted with vertices t3i20, then 1/2m2−2m2−1 distinct homomorphic copies μi19,i20′ of ψ are obtained. Furthermore, there are 6m2−2m2−1 pairs of vertices for these homomorphic copies.Table19 provides all information regarding μi19,i20′.Recall thatε2=0ifm3is even1ifm3is odd and let i21=1,2,3,…,m3−2+ε2/2 and i22=i21+1,i21+2,i21+3,…,m3−i21. The next theorem is obtained by contracting u3i21+1 with u3i22Table 19
Necessary information regarding homomorphic copiesμi19,i20′.
The pairs that are contractedt3i19+1,t3i20Family of homomorphic copies evolvedD20=μi19,i20′D201/2m2−2m2−1Number of pairs for eachμi19,i20′6Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD206m2−2m2−1Theorem 23.
If verticesu3i21+1 are contracted with vertices u3i22, then 1/4m32−2m3+ε2 distinct homomorphic copies μi21,i22″ of ψ are obtained. Furthermore, there are 3m32−3m3+2 pairs of vertices for these homomorphic copies.All details of the generated homomorphic copies are given in Table20.Now suppose thati23=1,2,3,…,m1−ε3/2. The following results have been constructed by contracting s3i23+1 with s3i23.Table 20
Necessary information regarding homomorphic copiesμi21,i22″.
The pairs that are contractedu3i21+1,u3i22:i21=1,2,3,…,m3−2+ε2/2 and i22=i21+1,i21+2,i21+3,…,m3−i21Family of homomorphic copies evolvedD21=μi21,i22″D211/4m32−2m3+ε2Number of pairs for eachμi21,i22″6Homomorphic copies possessing vertical axis of symmetry1/2m3−2+ε2Total number of pairs forD213m32−3m3+2Theorem 24.
If verticess3i23+1 are contracted with vertices s3i23, then 1/2m1−ε3 distinct homomorphic copies νi23,i23 of ψ are obtained. Furthermore, there are 3m1−1 pairs of vertices for these homomorphic copies.See Table21 for complete details of the evolved homomorphic copies.Now, we contractt3i24+1 with t3i24, where i24=1,2,3,…,m2−1.Table 21
Necessary information regarding homomorphic copiesνi23,i23.
The pairs that are contracteds3i23+1,s3i23Family of homomorphic copies evolvedD22=νi23,i23D221/2m1−ε3Number of pairs for eachνi23,i233Homomorphic copies possessing vertical axis of symmetryνm1/2,m1/2Total number of pairs forD223m1−1Theorem 25.
If verticest3i24+1 are contracted with vertices t3i24, then m2−1 distinct homomorphic copies νi24,i24′ of ψ are obtained. Furthermore, there are 6m2−1 pairs of vertices for these homomorphic copies.The complete information of the family of homomorphic copiesνi24,i24′ obtained in Theorem 25 is given in Table 22.Suppose thati25=1,2,3,…,m3−ε2/2. Let us contract u3i25+1 and u3i25 and obtain the following results.Table 22
Necessary information regarding homomorphic copiesνi24,i24′.
The pairs that are contractedt3i24+1,t3i24Family of homomorphic copies evolvedD23=νi24,i24′D23m2−1Number of pairs for eachνi24,i24′3Homomorphic copies possessing vertical axis of symmetryNo oneTotal number of pairs forD236m2−1Theorem 26.
If verticesu3i25+1 are contracted with vertices u3i25, then 1/2m3−ε2 distinct homomorphic copies νi25,i25″ of ψ are evolved. Furthermore, there are 3m3−1 pairs of vertices for these homomorphic copies.All information of the family of homomorphic copiesνi25,i25″ evolved in Theorem 26 is provided in Table 23.Lastly, we contract 4 pairs of vertices to evolve 4 homomorphic copies.Table 23
Necessary information regarding homomorphic copiesνi25,i25″.
The pairs that are contractedu3i25+1,u3i25Family of homomorphic copies evolvedD24=νi25,i25″D241/2m3−ε2Number of pairs for eachνi25,i25″3Homomorphic copies possessing vertical axis of symmetryνm3/2,m3/2″Total number of pairs forD243m3−1Theorem 27.
(i)
Letη1 be the homomorphic copy of ψ obtained by contracting s1 and t1. Then there are 6m2+m3+2 pairs of vertices for η1.(ii)
Letη2 be the homomorphic copy obtained by contracting s1 and u1. Then there are 3m2+2m3+2 pairs of vertices for η2.(iii))
Letη3 be the homomorphic copy obtained by contracting s1 and v1. Then there are 6m2+1 pairs of vertices to generate η3.(iv)
Letη4 be the homomorphic copy obtained by contracting t1 and u1.Then there are 6m3+1 pairs of vertices for η4.
LetΓ=0ifm1andm3are odd4ifm1andm3are even−1ifm1is even andm3is odd or vice versa. We are now in a position to prove the main theorem of this paper.Theorem 28.
There are1/4m12+4m22+m32+4m1m2+m2m3+2m1m2+8m2+4m3+Γ numbers of distinct homomorphic copies of ψ by contracting all the pairs in ψ.Proof.
Let us contract the following pairs of vertices:(i)
s3m1 and s3i1+1(ii)
t3m2 and t3i2+1(iii)
u3m3 and u3i3+1(iv)
s3m1 and t3i1+1(v)
s3m1 and u3i3+1(vi)
s3m1 and v3i4+1(vii)
t3m2 and s3i5+1(viii)
t3m2 and u3i6+1(ix)
u3m3 and t3i1+1(x)
v3m4 and u3i6+1(xi)
s3m1 and s3i7+1(xii)
v3m4 and s3i8+1(xiii)
v3m4 and t3i9+1(xiv)
s3i10+1 and t3i11(xv)
s3i10+1 and u3i12(xvi)
s3i13+1 and u3m3−ε2/2(xvii)
t3i14+1 and u3i15(xviii)
t3i14+1 and v3i16(xix)
s3i17+1 and s3i18(xx)
t3i19+1 and t3i20(xxi)
u3i21+1 and u3i22(xxii)
s3i23+1 and s3i23(xxiii)
t3i24+1 and t3i24(xxiv)
u3i25+1 and u3i25(xxv)
s1 and t1(xxvi)
s1 and u1(xxvii)
s1 and v1(xxviii)
t1 and u1
Then, by Theorems3 to 27, we obtain the set(4)F=αi1,αi2′,αi3″,βi1,βi3′,βi4″,βi5″′,βi6iv,βi1v,βi6vi,γi7,γi8′,γi9″,δi10,i11,δi10,i12′,δi13,m3−i3/2″,δi14,i15″,δi14,i16iv,μi17,i18,μi19,i20′,μi21,i22″,νi23,i23,νi24,i24′,νi25,i25″,η1,η2,η3,η4,of homomorphic copies of ψ and there are(5)S=3m22+3m2−2+3m22+3m2−4+3m32+3m3−2+3m22+3m2+3m32+3m3+32m42+3m4+32m12+3m1−4+3m32+3m3−4+32m22+3m2+32m32+3m3−4+6m2+2m1−m2−1+3m3+2m1−m3−1+6m3+2m2−m3−1+12m1−1m2−1+6m1−1m3−ε2−2+6m1−1ifm3is even12m1−1ifm3is odd+12m2−1m3−1+6m2−12+3m12+3m1−2+6m2−2m2−1+3m32−3m3−2+3m1−1+6m3−1+3m3−1+6m2+m3+2+3m2+2m3+2+6m2+1+6m3−1,pairs to form F. S=3m1+m2+m3+m42 is the total number of pairs in ψ and(6)F=14m12+4m22+m32+4m1m2+m2m3+2m1m2+8m2+4m3+Γ.
Thus, there are(7)14m12+4m22+m32+4m1m2+m2m3+2m1m2+8m2+4m3+Γ,numbers of distinct homomorphic copies obtained by contracting all pairs in ψ.


## 4. Conclusion

In this paper, we have highlighted the significance of homomorphic copies of coset graphs. We show how the bigger coset graphs get contracted to transform into their homomorphic copies. As a sample, we consider the closed pathψ of rank 4; that is, ψ=m1,m2,m3,m4, where m1>m2>m3 and m2=m4, in coset graphs. We prove that the total numbers of homomorphic copies of ψ are(8)14m12+4m22+m32+4m1m2+m2m3+2m1m2+8m2+4m3+Γ.It is hoped that the ideas presented in this study will also help to generate the homomorphic copies of other types of graphs for various purposes.



---

*Source: 3905425-2021-09-21.xml*