Inverse graphs associated with finite groups

Abstract

Let (Γ, * ) be a finite group and S a possibly empty subset of Γ containing its non-self-invertible elements. In this paper, we introduce the inverse graph associated with Γ 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. We then investigate its algebraic and combinatorial structures.

On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

“…Theorem 1 (see [10]). If Γ is a finite abelian group, which contains three or more elements, and if S is a nonempty subset of the non self-invertible elements, then GS(Γ) is a connected graph.…”

“…e radio number of G is the index of a radio labeling of G and is denoted by r n (G). Definition 4 (see [10]). Let (Γ, * ) be a finite group and…”

“…Since Cayley graphs, many relations between groups and graphs have been established and interest began to be on such relations. e inverse graphs associated with finite groups were recently introduced and analyzed by Alfuraidan and Zakariya [10], as a new example of such relations. ey developed some important graph theoretical aspects of the inverse graphs of certain finite groups that really shine as a spotlight on algebraic aspects of the groups.…”

Energy of inverse graphs of dihedral and symmetric groups

“…), began to gain the attention of researchers. Recently, Alfuraidan and Zakariya [1] introduced and studied the inverse graphs associated with finite groups. They established some interesting graph-theoretic properties of the inverse graphs of some finite groups which further shed more light on the algebraic properties of the groups.…”

“…In this paper, motivated by the works of Alfuraidan and Zakariya [1], Gutman [7], and Fadzil et al [9], we compute the energy of some inverse graphs which is the sum of the absolute values of the eigenvalues of adjacency matrices of their corresponding inverse graphs of finite groups. We study the energy of dihedral and symmetry groups and show that if G is a finite non-abelian group with exactly two non-self invertible elements, then the associated inverse graph (G) is never a complete bipartite graph.…”