On the inverse graph of a finite group and its rainbow connection number - Rian Febrian Umbara; A.N.M. Salman; Pritta Etriana Putri (Electronic Journal of Graph Theory and Applications (EJGTA), 2023)
A rainbow path in an edge-colored graph G is a path that every two edges have different colors. The minimum number of colors needed to color the edges of G such that every two distinct vertices are connected by a rainbow path is called the rainbow connection number of G. Let (Γ, *) be a finite group with TΓ = {t ∈ Γ|t ≠ t−1}. The inverse graph of Γ, denoted by IG(Γ), is a graph whose vertex set is Γ and two distinct vertices, u and v, are adjacent if u * v ∈ TΓ or v * u ∈ TΓ. In this paper, we determine the necessary and sufficient conditions for the inverse graph of a finite group to be connected. We show that the inverse graph of a finite group is connected if and only if the group has a set of generators whose all elements are non-self-invertible. We also determine the rainbow connection numbers of the inverse graphs of finite groups.