On Line Graph Associated with a Non-Commuting Graph for Finite Rings

Abstract

Let R be a non-commutative ring. A non-commuting graph of R, denoted by Γ_R, is a simple graph with a vertex set consisting of elements in R, except for its center. Any two distinct vertices x and y are adjacent if xyyx. A line graph associated with Γ_R, denoted by L(Γ_R), is a simple graph in which each vertex of L(Γ_R) represents an edge of Γ_R, and two distinct vertices of L(Γ_R) are adjacent if their corresponding edges share a common endpoint in Γ_R. This paper provides bounds for seven graph parameters of L(Γ_R): minimum degree, maximum degree, order, size, diameter, vertex-connectivity and edge-connectivity. Additionally, we show that the girth of L(Γ_R) is exactly 3.