On the inverse graph of a finite group and its rainbow connection number

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 T Γ = { t ∈ Γ | t ≠ t −1 } . The inverse graph of Γ , denoted by I G ( Γ ) , is a graph whose vertex set is Γ and two distinct vertices, u and v , are adjacent if u * v ∈ T Γ or v * u ∈ T Γ . 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.