Inverse Graphs Associated with Finite Groups

Let $(\Gamma,*)$ be a finite group and $S$ a possibly empty subset of $\Gamma$ containing its non-self-invertible elements. In this paper, we introduce the inverse graph associated with $\Gamma$ whose set of vertices coincides with $\Gamma$ such that two distinct vertices $u$ and $v$ are adjacent if and only if either $u * v\in S$ or $v * u\in S$. We then investigate its algebraic and combinatorial structures.


Introduction
Studying the algebraic structures of groups and rings based on graph combinatorial properties has caught the interests of many researchers over the last decades. Recently, graphs associated with rings seems to be more interesting and active area compare to those associated with groups. For instance, unit graphs [3], total graphs [1], zero divisor graphs [2], and co-maximal graphs [15] are few examples of graphs associated with rings. On the contrary, the main aim of this paper is to introduce inverse graph and establish some of its properties. It is hoped that this work will shed more light on the algebraic structures of groups via graphs and open doors for more researches in this approach.
There has been a close tie between graph and group theories for more than a century. Deep investigations and amazing results from group theory have been proved easily via combinatorial properties of graphs and vice versa, [4,8]. This serves as the underlying motivation for this work. It was Arthur Cayley in [7] who was the first to associate graphs "Cayley graphs" to finite groups. Given a finite group Γ and a non-empty generating subset C ⊆ Γ, the Cayley graph denoted by Cay(Γ, C) is defined as follows: each vertex is an element of Γ with two vertices u, v ∈ V (Cay(Γ, C)) being adjacent if uv −1 ∈ C, see [14]. The connectivity and the planarity of this graph was investigated in [10]. For an excellent survey on the Hamiltonian cycles in Cayley graphs see [17]. Aside this, several other graphs have also been associated with finite groups such as commuting graphs, intersection graphs, prime graphs, non-commuting graphs, conjugacy class graphs, etc, [5,6,11,12,18]. In order to justify our claim that the inverse graph is new, we as well illustrate by examples how it is different from some known graphs associated with groups.

Preliminaries
A graph G is formed by a finite non-empty set V (G) of elements called vertices together with a possibly empty set E(G) of 2−element subsets of V (G) called edges. The cardinality of V (G) is called the order of G, while the cardinality of E(G) is called the size of G. Two vertices u and v of G are adjacent if there is an edge between them i.e. {u, v} ∈ E(G). In this case, we write uv ∈ E(G). A vertex u is said to be incident to an edge if u is one of the two vertices of G that form the edge. The number of edges incident to a vertex u in G is the degree of u, denoted by deg(u). A graph G with vertices each of degree r is called an r−regular graph. Two or more edges that join the same pair of distinct vertices are called parallel edges. A loop is an edge that joins a vertex to itself. A walk of length k ≤ n in a graph G with vertex set whose terms alternate between vertices and edges such that v i t−1 v it = a jt for 1 ≤ t ≤ k and open. An open walk in which no vertex is repeated is called a path. A closed walk of length k ≥ 3 with neither vertex nor edge is repeated is called a cycle. A vertex v of a graph G is said to be reachable if there is a walk starting from or ending at v from every other vertex in G. If every vertex in G is reachable and G is non-empty 1 then G is connected otherwise it is disconnected.
Definition 2.1. Let (Γ, * ) be a finite group and S = {u ∈ Γ | u = u −1 }. We define the inverse graph G S (Γ) associated with Γ as the graph whose set of vertices coincides with Γ such that two distinct vertices u and v are adjacent if and only if either u * v ∈ S or v * u ∈ S.
Throughout this paper S will always denote the non-self-invertible elements of the group Γ.
Remark 2.1. It is worthwhile to observe the following: 1. Clearly, identity e is a trivial self-invertible element in a finite group Γ. Hence e / ∈ S. Consequently, the cardinality of S is strictly less than the cardinality of Γ. In particular, if Γ contains no self-invertible element other than the identity then |S| = |Γ| − 1.

Properties of the Inverse Graphs
In this section, we study some basic properties of inverse graphs associated with finite groups. Proof. Suppose G S (Γ) is empty. Then it follows from Definition 3.1 that for any u, v ∈ Γ with u = v, we have u * v / ∈ S. Hence S is empty i.e., |S| = 0. The converse is obvious.
Remark 3.1. Proposition 3.1 characterizes the inverse graph of a finite group with all of its elements being self-invertible. In particular, the inverse graphs associated with the following groups are empty graphs: 1. Groups consisting of two elements.
, with usual addition of matrices.
On the other extreme, it is natural to ask when will the inverse graph be complete? Unfortunately, such graph never exists. A graph G is said to be complete provided it is non-empty and there is an edge between every distinct pair of its vertices. In a complete graph G with |G| = n, we have deg(v) = n − 1 for any vertex v of G.
Theorem 3.1. There is no inverse graph that is complete for any finite group Γ.
Proof. Suppose on the contrary that there exists an inverse graph G S (Γ) that is complete. Then for Case I: n is even. Since deg(e) = n − 1, then by Remark 2.1-(3) we have |S| = n − 1 which is not possible as |S| is always even for any inverse graph.
Since there is an edge between every two distinct vertices, we have e = u * v ∈ S which is not possible, see Remark 2.1-(1).
Theorem 3.2. For any finite group Γ with at least three elements and a non-empty subset S of non-self-invertible elements, the graph G S (Γ) has no isolated vertex.
Proof. Suppose by contradiction that there exists an isolated vertex v in G S (Γ). Then we have the following two cases.
, there is an edge between v and v ′ , a contradiction.
Since both u and v are arbitrarily chosen we have every element of Γ \ {S ∪ {e}} to be adjacent to each element of S. Therefore, each vertex of G S (Γ) is reachable and therefore connected.
Remark 3.2. Note that the commutativity of Γ in Theorem 3.3 can not be dropped for the conclusion to hold. An example of a non-abelian group with a disconnected inverse graph is the symmetry group S 3 of order 6. On the other hand, it is not true that the inverse graph of every non-abelian group is disconnected. A counterexample is the inverse graph associated with quaternion group Q 8 .
In a connected graph G, the distance between two vertices u, v ∈ V (G), is the length of the shortest path between u and v. It is denoted by d(u, v). Eccentricity ρ(u), of a vertex u in a connected graph G is defined as The minimum and the maximum eccentricity in a graph are radius, rad (G), and diameter, diam(G), of G respectively. Proof. Let G S (Γ) be a connected inverse graph associated with a group Γ. We consider the following vertex partition: V (G S (Γ)) = {e} ∪ S ∪ S ′ , where S is the set of all non-self-invertible elements and S ′ is the set of all non-trivial self-invertible elements (called the set of involutions). Since every element in S is adjacent to e and there is no edge between e and the elements of S ′ , we have the eccentricity of e, ρ(e) = 2. Also, for each element u ∈ S, ρ(u) = 2 as u is not adjacent to its inverse but adjacent to e and every element in S ′ . Now take an arbitrary element v ∈ S ′ , it follows from the construction and connectivity of G S (Γ) that v is not adjacent to e but adjacent to every vertex in S. Hence ρ(v) = 2.
The following Lemma is the inverse graph version of the first theorem of graph theory. Proof. Let V (G S (Γ)) = {v 1 , v 2 , · · · , v n }. By the first theorem of graph theory, we have Since G S (Γ) is a non-empty graph, then by Proposition 3.1 S = ∅. As any pair u, v ∈ S with u = v −1 has no edge in G S (Γ), such a pair contributes −2 to the total degrees of the vertices of G S (Γ). Hence a total of −|S| degrees will be contributed by the elements of S as a shortage in the sum of the degrees of G S (Γ). Therefore, The following corollary elicits the fact that the inequality in Lemma 3.1 cannot be improved. Proof. Let V (G S (Γ)) = {v 1 , v 2 , · · · , v n }. Without loss of generality, let v 1 = e, the identity element of Γ. By remark 2.1-(1), we have |S| = |Γ| − 1 = n − 1, i.e., S = {v 2 , . . . , v n }. Hence, Example 3.1. Let Γ = {z ∈ C : z 5 = 1} be the group of fifth root of unity under multiplication. As Γ is generated by one of its elements say ω, then we can write Γ =< ω >= {ω, ω 2 , · · · , ω 5 = 1}. Figure 2.  . Now, suppose that Γ 1 = ω = e, ω, . . . , ω n−1 is a cyclic group of order n. As n is odd, then ω i ∈ Γ 1 is non-self-invertible since (ω i ) −1 = ω j where i + j = n. It follows from Corollary 3.1 that the size of G(Γ 1 ) is equal to n(n−1)−|S| 2 .

Theorem 3.5. Let Γ be a finite group and the set S of non-self-invertible elements be a non-empty. The associated inverse graph G S (Γ) is (i) 2−regular if Γ is a group of four elements.
(ii) (2 n − 2)−regular if Γ is a generalized quaternion group of order 2 n where n > 2.
Since S is non-empty, without lost of generality, S = {v 1 , v 2 }. By construction G S (Γ) is the cycle C 4 . Hence it is 2 − regular.
Then there exists a generalized quaternion group of order 2 n such that its inverse graph is isomorphic to G.
A graph G is bipartite if V (G) can be partitioned into two disjoint sets V 1 and V 2 such that uv ∈ E(G) implies either u ∈ V 1 and v ∈ V 2 or v ∈ V 1 and u ∈ V 2 . If each vertex in V 1 is adjacent to each vertex in V 2 , then G is a complete bipartite graph. Suppose |V 1 | = r and |V 2 | = s, then this complete bipartite graph, denoted by K s,r , has order s + r and size sr . (i) u ∈ V 1 and v ∈ V 2 . If u * v = v 0 ∈ V 2 , then as Γ is abelian, we have e = v 2 0 = u 2 * v 2 = u 2 * e = u 2 which implies that u = u −1 . But this is not possible as u ∈ V 1 . Hence u * v ∈ V 1 . Therefore, there is an edge between them.
(ii) u, v ∈ V 1 . As there are only two elements in V 1 , then u * v = e ∈ V 2 . Therefore, there is no edge between u and v.
Therefore, there is no edge between u and v.
A graph G is planar provided it can be represented on a plane such that there is no crossing of its edges. It is a fact that every complete bipartite graph K 2,r is planar, 2−colourable and has a clique number of 2. Hence the following corollary is obvious. We conclude this section with the following two results that classify the inverse graphs of Z 4 . Proof. Let Γ be a group of four elements and S be the set of non-self-invertible elements of Γ. Since elements of S occur in pairs, |S| is equal to 0, 2, or 4. |S| cannot be zero because this would imply that G S (Γ) is empty. Also |S| = 4 since e / ∈ S. Hence |S| = 2 and Γ has exactly one non-trivial self-invertible element. Proof. Suppose that G S (Γ) = C 4 . By contrary, assume that Γ ≇ Z 4 . Since G S (Γ) is a non-empty graph, then by Lemma 3.2, we have n = |Γ| = 4.
(i) n = 3. Not possible as the only non-empty inverse graph of order 3 is P 2 , the path of length 2.
(ii) n ≥ 5. As each vertex of G S (Γ) has degree two, then by Remark 2.1-(3), 2 = deg(e) = |S|. Therefore, exactly two elements u and v ∈ Γ are non-self-invertibles. Since e is adjacent to both of them in G S (Γ), there must exist a non-trivial self-invertible element ω that is adjacent to u. Hence u * ω = v which implies that v * ω = u. Thus ω is adjacent to both u and v, see Figure 3. But as |Γ| ≥ 5, we must then have G S (Γ) to be disconnected which is again not true as our inverse graph is C 4 .
Thus the only possible value of the order is four and hence by Lemma 3.2 Γ ∼ = (Z 4 , +).  The converse follows immediately by constructing G S (Z 4 ).
A graph G is said to be Hamiltonian provided G contains a cycle that visits all its vertices. It follows immediately that every cycle C n is Hamiltonian and hence G S (Z 4 ) is Hamiltonian. The following theorems give some Hamiltonian characterizations of inverse graphs. Proof. We have two cases n is even, odd. Case I: n is even. In this case, Z n has exactly one non-trivial self-invertible element, say u with deg(u) = deg(e) = n − 2. For each of the remaining non-self-invertible element v ∈ Z n , we have Therefore, deg(e) = deg(u) = n − 2 > n − 3 ≥ n 2 , for all n ≥ 6. Thus by Dirac's theorem the graph G S (Z n ) is Hamiltonian. Case II: n is odd. In this case, Z n has the identity e as the only self-invertible element. Therefore, deg(v i ) = n − 2 for all v i = e ∈ G S (Z n ) (see the proof of Corollary 3.1). By Remark 2.1-3, deg(e) = n − 1 > n − 2 ≥ n 2 , ∀n ≥ 6. Thus by Dirac's theorem the graph G S (Z n ) is Hamiltonian.
As a consequence of Theorems 3.8 and 3.9, we have the following corollaries.   Proof. Let Γ = {e} ∪ T ∪ S, where T is the set of all non-trivial self-invertible elements of Γ and S is the set of non-self-invertible elements in Γ. We have |T | = n 2 − 1 and |S| = n 2 . By and therefore, v ∈ T . So there is no edge between any pair of non-trivial self-invertible elements in G S (Γ). However, for v i ∈ T and v j ∈ S, we must have v i * v j ∈ S. Otherwise, there exists an element v k ∈ T such that v i * v j = v k which implies that v j = v i * v k ∈ T , a contradiction. Therefore, deg(v) = |S| = n 2 for each element v ∈ T . As for each v j ∈ S, elements of T contribute n 2 − 1 to its degree, identity contributes 1 and so we have deg(v i ) ≥ n 2 since there could be edges between any pair of elements of S. Thus by Dirac's theorem G S (Γ) is Hamiltonian.
The concept of isomorphism is of prime importance in both group and graph theories. Among the three crucial problems for groups raised by Max Dehn in 1911, the isomorphism problem is the most difficult (the word and conjugacy problems constitute the other two: see [13] and [9]). The problem of deciding whether two words (using some algorithms) in some finite generating sets represent the same or conjugate elements in a group constitutes the same word are the word and conjugacy problems respectively. Isomorphism problem, on the other hand, has to do with determining whether two groups that appear different are actually the same. In fact, Francois and Vincent asserted that the problem of finding an isomorphism of two groups is unsolvable for some classes of groups, [9]. This is quite interesting as a lot of researches are still going on to solve these problems.
Informally speaking, isomorphic groups are really the same groups except for the notations used. Besides, if Γ 1 and Γ 2 are two isomorphic groups then they share the same group -theoretic properties such as abelian, cyclic, elements of finite order, numbers of involutions, etc. It is clear from the definition of inverse graphs that if Γ 1 ∼ = Γ 2 then G S 1 (Γ 1 ) ∼ = G S 2 (Γ 2 ) where S 1 and S 2 are the sets of non-self-invertible elements in Γ 1 and Γ 2 respectively. Investigation into the converse of this statement constitutes the conclusion of this section.
As a consequence of Proposition 3.1 and Corollary 3.2 we have the following result.
(ii) A generalized quaternion group.