# Equal-Square Graphs Associated with Finite Groups

Authors: Shafiq Ur Rehman; Ghulam Farid; Tayaba Tariq; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2022)

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## Abstract

The graphical representation of finite groups is studied in this paper. For each finite group, a simple graph is associated for which the vertex set contains elements of group such that two distinct verticesx and y are adjacent iff x2=y2. We call this graph an equal-square graph of the finite group G, symbolized by ESG. Some interesting properties of ESG are studied. Moreover, examples of equal-square graphs of finite cyclic groups, groups of plane symmetries of regular polygons, group of units Un, and the finite abelian groups are constructed.


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## Body

## 1. Introduction

Graphs are studied very extensively by taking into account some characteristics and properties, which are used to prepare set of vertices and constructing edges. Almost all fields of sciences and social sciences are studied in the prospect of theory of graphs. Even the abstract areas of mathematics are linked with graphs. Discrete mathematics, combinatorics, and number theory play a very significant role in this study; also, concepts of abstract algebraic structures are studied with the help of graphs.Visualizing groups using graphs is a rapidly growing trend in algebraic graph theory [1–3]. The automorphism group of a graph and the Cayley graph of a group motivates to study the interplay between graphs and groups, see [4–8]. There exists numerous ways to associate a graph with certain groups by taking subgroups or elements as vertices, and two vertices are adjacent iff they satisfy a certain relation. Such graphs are characterized through the set of vertices and the adjacency relation between the vertices. In recent years, graph theory and its applications associated to algebraic structures are studied extensively. To contribute in such a commendable area, we introduce here a new type of combination. We construct a graph whose vertex set is a finite group G and has two distinct vertices a and b adjacent iff a2=b2. We call this graph an equal-square graph of G and will be represented by ESG.A large amount of literature is devoted to study the graphs associated to finite groups, for instance, commuting graphs [9–15], noncommuting graphs [16, 17], intersection graphs [18], prime graphs [19–21], conjugacy class graphs [22], power graphs [23–25], inverse graphs [26], quadratic residues graphs [27], order divisor graphs [28], and square graphs [29].For reader’s facilitation, some basic terminologies are recalled here from [30–32]. The dihedral group, symbolized by Dn, is the group of plane symmetries of an n-sided regular polygon n≥3. This group has order 2n and is denoted by Dn=u,v|un=v2=uv2=e. The group of units of the ring ℤn is symbolized by Un, i.e., Un=y¯∈ℤn|y,n=1. For some fixed prime p, a group G is called a p-group if, for each x∈G, we have x=pk for some k∈ℤ+. For some fixed prime p, an abelian p-group of exponent p is called elementary abelian p-group. A finite elementary abelian p-group is isomorphic to ℤpk=ℤp⊕ℤp⊕⋯⊕ℤp (direct sum of k-copies), for some k∈ℤ+. Elementary abelian 2-groups are sometimes called Boolean groups, cf. [33].A graphG that contains only vertices without any edge is known as an empty graph. Two vertices having an edge or edges between them are termed as adjacent vertices. An undirected graph G that does not contain parallel edges, loops, or multiple edges is called a simple graph. A simple graph of order n in which every two vertices are adjacent to each other is called the complete graph, symbolized by Kn. A graph in which no edges cross each other is called a planar graph. If we can partition the vertex set of a graph G into two nonempty disjoint subsets such that each edge has one vertex in each partition, then the graph G is called a bipartite graph. The disjoint union of graphs G1 and G2 is the graph G1+G2 such that VG1+G2=VG1∪VG2 and EG1+G2=EG1∪EG2. The graph kG=G+G+⋯+G denotes the disjoint union of k copies of graph G.Our goal in this work is to associate a finite group with new graph representation. Corresponding to each finite groupG, we will denote by ESG, the equal-square graph of G, whose vertex set consists of all the elements of G such that two distinct vertices a and b are adjacent iff a2=b2 (Definition 1). Some examples in this regard are ESℤ2⊕ℤ2=K4 (Example 1) and ESD6=K8+2K2 (Example 3). In Section 2, we describe some basic properties of ESG and investigate that when ESG is connected or complete. We obtain the following results in this section: G is odd iff ESG is empty (Theorem 1). ESG is connected iff ESG is complete iff G is elementary abelian 2-group (Theorem 2). We conclude that, for every finite group G, ESG is either a complete graph or it is a disjoint union of complete graphs (Remark 2). In Section 3, we completely describe the equal-square graph of cyclic groups. We obtain the following results in this section: ESG=nK2, for every cyclic group G of order 2n (Theorem 3). However, in general, the converse is not true (Remark 3). For every prime p≠2, ESFpn∗=pn−1/2K2 (Corollary 1). In Section 4 we describe the equal-square graph of the dihedral group. We prove that ESDn=Kn+1+n−1K1 if n is odd and ESDn=Kn+2+n−2/2K2 if n is even (Theorem 4). In Section 5, we characterize the equal-square graph of the group of units Un (Theorems 5–7).In Section6, we completely describe the equal-square graph of the finite abelian groups. Mainly, we obtain the following results in this section. If Gi=ai are cyclic groups of order ni, for each 1≤i≤k, then ES⊕i=1kGi=n1n2⋯nk/2rK2r, where r denotes the number of even integers in the set n1,n2,…,nk (Theorem 9). If G≅ℤ2β1⊕⋯⊕ℤ2βt⊕ℤp1β11⊕⋯⊕ℤp1β1t1⊕⋯⊕ℤpkβk1⊕⋯⊕ℤpkβktk, where β1≥β2≥⋯≥βt≥1 with β1+β2+⋯+βt=α and βi1≥βi2≥⋯≥βiti≥1 with βi1+βi2+⋯+βiti=αi, for each i; 1≤i≤k, then ESG=G/2tK2t (Theorem 10). If G is an abelian group of finite order G=2αλ, provided λ is odd and α≥1, then we will have ESG=G/2tK2t for each partition α=d1+d2+⋯+dt of the integer α of length t (Corollary 2). If G is an abelian group of finite order G=2αλ, provided λ is odd and α≥1, then G can have α distinct (nonisomorphic) equal-square graphs (Corollary 3). All possible equal-square graphs of an abelian group of order 36000 are described (Example 6 and Table 1). The equal-square graphs of small order groups are described at the end (Example 7 and Table 2).Table 1
Equal-square graph of abelian group of order 36000.
Partitions ofα=5Length of partitionsEqual-square graph1+1+1+1+1536000/25K25=1125K321+1+1+2436000/24K24=2250K161+1+3,2+2+1336000/23K23=4500K81+4, 2 + 3236000/22K22=9000K45136000/2K2=18000K2Table 2
Equal-square graphs of small-order groups.
OrderGroupsEqual-square graphs1eK12ℤ2K23ℤ33K14ℤ4,ℤ2⊕ℤ22K2,K45ℤ55K16ℤ6,S33K2,K4+2K17ℤ77K18ℤ2⊕ℤ2⊕ℤ2,ℤ2⊕ℤ4,ℤ8,D4,Q8K8,2K4,4K2,K6+K29ℤ9,ℤ3⊕ℤ39K110ℤ10,D55K2,K6+4K1Throughout this paper, all the groups will be finite. Moreover, all the graphs will be finite and simple. Any unexplained material will be standard as in [30–32].


## 2. Properties ofESG

This section presents some basic properties ofESG. We determine that the groups with odd order have empty equal-square graphs. Furthermore, we investigate that when ESG is connected or complete. We start by introducing the equal-square graph ESG and some of its examples.Definition 1.
LetG be a finite group with group operation written multiplicatively. A graph will be called an equal-square graph if its vertex set consists of entire elements of G and has two distinct The equal-square graph ESℤ2⊕ℤ2 of the Klein group ℤ2⊕ℤ2-square graph of the group G.Example 1.
The equal-square graphESℤ2⊕ℤ2 of the Klein−4 group ℤ2⊕ℤ2 is presented in Figure 1.Figure 1
ESℤ2⊕ℤ2.Example 2.
The dihedral groupD6=a,b|a6=b2=ab2=e of order 12 is obtained by the symmetries of a regular hexagon in the plane. The equal-square graph ESD6 of D6 is as in Figure 2.Figure 2
ESD6.Example 3.
The dihedral groupD4=a,b|a4=b2=ab2=e (group of symmetries of a square in plane) and the group of quaternions Q8 have the same equal-square graph, as shown in Figure 3.Figure 3
ESD4≅ESQ8.Theorem 1.
A groupG has odd order iff ESG is empty.Proof.
SupposeG has odd order, and let x2=y2, for some x,y∈G. Then, x=m1 and y=m2, for some odd positive integers m1 and m2. We can assume that 2≤m1≤m2. Now, x2=y2 implies that x2m2=y2m2=e, and hence, m1|2m2. Since m2 is odd, so m1|m2. Similarly, m2|m1, and hence, m1=m2. We can write m1=m2=2k+1 for some positive integer k. However, then, x=xe=xx2k+1=x2k+1=y2k+1=y2k+1y=ey=y. Hence, ESG is empty.
Conversely, supposeESG is empty. If G is even, Cauchy’s theorem ensures that G has an element of order 2 which is adjacent to e, a contradiction.Remark 1.
We have seen in the above result that the groups having odd order have an empty equal-square graph. Therefore, we are mostly interested in the equal-square graph of the groups having even order.Theorem 2.
The following assertions are equivalent for a groupG with G≥2.(a)
ESG is connected(b)
ESG is complete(c)
G is elementary abelian 2-groupProof.
(a)⇒ (b): suppose ESG is connected. Then, for each a∈G, there exist a1,a2,,,,an∈G such that a−a1−a2−⋯−an−e. This implies that a2=a12=a22=⋯=e2=e. We conclude that a2=e, for each a∈G, and hence, x2=y2, for all x,y∈G. It follows completeness of ESG.
(b)⇒ (c): if ESG is complete, then x2=y2, for all x,y∈G. In particular, x2=e, for all x∈G. This implies that each nonidentity element in G has order 2. This implies that G is abelian and hence an elementary abelian 2-group.
(c)⇒ (a): G is elementary abelian 2-group. Then, each nonidentity element in G has order 2. This implies that x2=e, for all x∈G. Hence, each vertex in G is adjacent to e and so ESG is connected.Example 4.
The equal-square graph of an elementary abelian 2-group is given by(1)ESℤ2⊕ℤ2⊕⋯⊕ℤ2︸m−times=K2m.Remark 2.
LetG be a finite group. If we define G2=x2|x∈G, then clearly 1≤G2≤G. Moreover, if G2=1, i.e., G2=e, then ESG is a complete graph. If G2=2, then ESG is a disjoint union of two complete graphs. If G2=3, then ESG is a disjoint union of three complete graphs and so on. If G2=G, then ESG is empty, i.e., ESG=GK1. We conclude that, for a finite group G, ESG is either a complete graph or it is a disjoint union of complete graphs.


## 3. Equal-Square Graph of Cyclic Groups

In this section, we completely describe the equal-square graph of cyclic groups. Recall thatG1+G2 denotes the disjoint union of graphs G1 and G2. Moreover, kG=G+G+⋯+G denotes the disjoint union of k copies of graph G.Theorem 3.
For a cyclic groupG of order 2n, ESG=nK2.Proof.
LetG=a=e,a,a2,a3,…,an,…,a2n−1. Then, ai2=an+i2, where 0≤i≤n−1. Now, let ai be adjacent to aj such that 0≤i≤j≤2n−1, then ai2=aj2⇒a2i=a2j⇒a2j−i=e⇒a divides 2j−i⇒2n|2j−i⇒n|j−i,0≤j−i≤2n−1⇒j−i=n or 0, then j=i+n or j=i. Therefore, each ai is adjacent to only one vertex, i.e., an+i, where 0≤i≤n−1. Hence, ESG=nK2.Remark 3.
The groupℤ3⊕ℤ6 is not cyclic, but ESℤ3⊕ℤ6=9K2. This implies that the converse of the above statement in general does not hold.Corollary 1.
Ifp≠2 is prime and Fpn denotes the finite field of order pn, then(2)ESFpn∗=pn−12K2.Proof.
Apply Theorem3 and the fact that the multiplicative group Fpn∗ of every finite field Fpn is a cyclic group, cf. Theorem 22.2 of [31].


## 4. Equal-Square Graph of Dihedral Groups

In this section, we completely describe the equal-square graph of the group of plane symmetries ofn-gon n≥3.Theorem 4.
The equal-square graph of the dihedral groupDn=α,β|αn=β2=αβ2=e is given by(3)ESDn=Kn+2+n−22K2,ifniseven,Kn+1+n−1K1,ifnisodd.Proof.
Supposen is even. Let ak=2, for some k, where 1≤k≤n−1. Then, n/n,k=2 implies that k=n/2. Also, aib=2, for all i, 0≤i≤n−1. Thus, we have total n+1 elements of order 2, namely, an/2,b,ab,a2b,,a3b,…,an−1b. Hence, by joining all these to identity, we get a complete subgraph Kn+2 in ESDn. Now, we are left with n−2 elements, namely, a1,a2,…,an/2−1,an/2+1,…,an−1 in Dn. Now, by taking squares of all these remaining elements, we see that a is adjacent to an/2−1, a2 is adjacent to an/2+1, a3 is adjacent to an/2+3, and so on. In this way, we get disjoint union of n−2/2 copies of K2 in ESDn. Hence, ESDn=Kn+2+n−2/2K2.
Now, suppose thatn is odd. Then, no power of a has order two. Thus, the only elements of order two will be b,ab,a2b,,a3b,…,an−1b, which by joining to identity, form a complete subgraph Kn+1 in ESDn. Then, we are left with n−1 elements, namely, a1,a2,…,an−1 in Dn. Suppose ai2=aj2, for some i≠j, with 1≤i,j≤n−1. Then, n|2i−j. Since n is odd, so n|i−j. This implies that ai−j=e, and thus, ai=aj. Hence, ESDn=Kn+1+n−1K1.


## 5. Equal-Square Graph of the Group of UnitsUn

This section completely describes the equal-square graph of the group of unit elements of the ringℤn. By using the results proved in [27], we can completely describe the equal-square graph of the group of unit elements of the ring ℤn as follows.Theorem 5.
LetUn be the group of units of the ring ℤn.(a)
ESU2 is the empty graph(b)
ESU22=K2(c)
ESU2s=2s−3K4, for each integer s≥3(d)
ESUp=p−1/2K2, for each prime p≠2(e)
ESUps=ps−1p−1/2K2, for each prime p≠2 and for each positive integer sProof.
Apply Theorem 2.1 of [27].Theorem 6.
Letn=p1α1⋅p2α2⋯pmαm; pis are distinct odd primes and αis∈ℤ+. Then,(4)ESUn=ϕn2mK2m.Proof.
Apply Theorem 2.2 of [27].Theorem 7.
Letn=2s⋅p1α1⋅p2α2⋯pmαm; pis≠2 are distinct primes and αis≥1, s≥0 are integers. Then,(5)ESUn=ϕn2mK2m,ifs=0or1,ϕn2m+1K2m+1,ifs=2,ϕn2m+2K2m+2,ifs≥3.Proof.
Apply Theorem 2.3 of [27].


## 6. Equal-Square Graph of Finite Abelian Groups

In this section, we completely describe the equal-square graph of the finite abelian groups. LetΓ be the collection of all finite simple graphs (including the empty graph). First, we recall some definitions and results from [34].Some basic properties of the strong product are recalled below.Proposition 1 (see Sections 4.1, 4.2, and 5.2 of [34]).
Let “+” and “⊠” be the operations of disjoint union and strong product defined on Γ, respectively. The following assertions hold.(a)
G1⊠G2≅G2⊠G1, for all G1,G2∈Γ, i.e., ⊠ is commutative(b)
G1⊠G2⊠G3≅G1⊠G2⊠G3, for all G1,G2,G3∈Γ, i.e., ⊠ is associative(c)
G1⊠G2+G3=G1⊠G2+G1⊠G3, for all G1,G2,G3∈Γ, i.e., ⊠ is distributive over +(d)
K1⊠G≅G, for all G∈Γ, i.e., the trivial graph K1 is a unity in ΓDefinition 2.
(see Sections 4.2 and 5.2 of [34]). For graphs G1,G2,G3,…,Gk in Γ, the strong product ⊠i=1kGi=G1⊠G2⊠⋯⊠Gk∈Γ is the graph whose vertex set is the Cartesian product VG1×VG2×⋯×VGk such that two distinct vertices a1,a2,…,ak and b1,b2,…,bk are adjacent iff aibi∈EGi or ai=bi, for each i; 1≤i≤k. The kth power of a graph G∈Γ with respect to ⊠ is symbolized by G⊠,k and is defined as G⊠,k=G⊠G⊠⋯⊠G︸k−times.Lemma 1.
rKm⊠sKn=rsKmn, for all m,n,r,s∈ℤ+.Proof.
We haveKm⊠Kn=Kmn, see Exercise 4.7 of [34]. Therefore,(6)rKm⊠sKn=Km+Km+⋯+Km︸r−times⊠Kn+Kn+⋯+Kn︸s−times=Km⊠Kn+Km⊠Kn+⋯+Km⊠Kn︸rs−times=Kmn+Kmn+⋯+Kmn︸rs−times=rsKmn.Theorem 8.
For any two groupsG and H in Γ, ESG⊕H=ESG⊠ESH.Proof.
Clearly, we have thatVESG⊕H=VESG⊠ESH=G⊕H. Let a,b,a′,b′∈G⊕H such that a,b,a′,b′∈EESG⊕H. Then, a,b2=a′,b′2, and hence, a2=a′2 and b2=b′2. From a2=a′2, we get that either a=a′ or aa′∈EG. Also, from b2=b′2, we get that either b=b′ or bb′∈EH. Combining, we obtain that a=a′ and bb′∈EH, or aa′∈EG and b=b′, or aa′∈EG and bb′∈EH. Hence, a,ba′,b′∈EESG⊠ESH.
Conversely, we suppose thata,b,a′,b′∈G⊕H such that a,ba′,b′∈EESG⊠ESH. This implies that a=a′ and bb′∈EH, or aa′∈EG and b=b′, or aa′∈EG and bb′∈EH. From this, we conclude that either a=a′ or a2=a′2 and also either b=b′ or b2=b′2. Therefore, a2=a′2 and b2=b′2, and hence, a,ba′,b′∈EESG⊕H.Theorem 9.
LetGi=ai be cyclic groups of order ni for each 1≤i≤k. Then,(7)ES⊕i=1kGi=n1n2⋯nk2rK2r,where r denotes the number of even integers in the set n1,n2,…,nk.Proof.
Applying Theorem8, we obtain(8)ES⊕i=1kGi=⊠i=1kESGi.
Also, by Theorems1 and 3, we have, for each 1≤i≤n,(9)ESGi=niK1,ifniisodd,ni2K2,ifniiseven.
Combining the above two equations and then by using Lemma1, we obtain(10)ES⊕i=1kGi=n1n2⋯nk2rK2r.Examples 5.
We can immediately determine the equal-square graph of the direct product of cyclic groups as follows:(11)ESℤ3⊕ℤ4⊕ℤ5⊕ℤ6=3⋅4⋅5⋅622K22=90K4,ESℤ2⊕ℤ6⊕ℤ8⊕ℤ9⊕ℤ12=2⋅6⋅8⋅9⋅1224K24=648K16,LetG be a finite abelian group with G=2αp1α1p2α2⋯pkαk, where p1,p2,…,pk are distinct odd primes and α,αis are positive integers. Then, G can be uniquely expressed as G≅ℤ2β1⊕⋯⊕ℤ2βt⊕ℤp1β11⊕⋯⊕ℤp1β1t1⊕⋯⊕ℤpkβk1⊕⋯⊕ℤpkβktk, where β1≥β2≥⋯≥βt≥1 with β1+β2+⋯+βt=α and βi1≥βi2≥⋯≥βiti≥1 with βi1+βi2+⋯+βiti=αi, for each i; 1≤i≤k, cf. Section 5.2, Theorem 5 [30].Theorem 10.
With notations above,ESG=G/2tK2t.Proof.
Applying Theorem8, we obtain that ESG=ESℤ2β1⊠⋯⊠ESℤ2βt⊠ESℤp1β11⊠⋯⊠ESℤp1β1t1⊠⋯⊠ESℤpkβk1⊠⋯⊠ESℤpkβktk. Then, by using Theorems 1 and 3, we get that ESG=2β1−1K2⊠⋯⊠2βt−1K2⊠p1β11K1⊠⋯⊠p1β1t1K1⊠⋯⊠pkβk1K1⊠⋯⊠pkβktkK1. Furthermore, by using Lemma 1, we get that ESG=2β1+⋯+βt−tp1α1⋯pkαkK2t=G/2tK2t.Corollary 2.
LetG be an abelian group with order G=2αλ, where λ is odd and α≥1. Then, we have the equation ESG=G/2tK2t for each partition α=d1+d2+⋯+dt of the integer α of length t.Proof.
We can writeλ=p1α1p2α2⋯pkαk, where p1,p2,…,pk are distinct odd primes and αis are nonnegative integers. In light of Theorem 10, for each partition α=d1+d2+⋯+dt of length t, we get that ESG=G/2tK2t.Corollary 3.
LetG be an abelian group with order G=2αλ, where λ is odd and α≥1. Then, it can have exactly α distinct (nonisomorphic) equal-square graphs.Example 6.
All possible equal-square graphs of an abelian group of order36000=253253 are described in Table 1.Example 7.
The equal-square graphs of small order groups are described in Table2.


## 7. Concluding Remarks

Graph theory is an interesting area of research. A lot of topics in different areas of research are beautifully presented in the form of graphs. This work relates the finite group structures with simple graphs in the form of equal-square graphs which is defined by a particular adjacency relation on vertices. We have presented several examples in the support of results proved in this paper. Also, we have constructed tables for different well-known groups of finite order which provide corresponding equal-square graphs.



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