On Line Graph Associated with a Non-Commuting Graph for Finite Rings - Natwalan Ruk-u; Khajee Jantarakhajorn; Borworn Khuhirun (Science & Technology Asia, 2023)
Let R be a non-commutative ring. A non-commuting graph of R, denoted by ΓR, 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 xy yx. A line graph associated with ΓR, denoted by L(ΓR), is a simple graph in which each vertex of L(ΓR) represents an edge of ΓR, and two distinct vertices of L(ΓR) are adjacent if their corresponding edges share a common endpoint in ΓR. This paper provides bounds for seven graph parameters of L(ΓR): minimum degree, maximum degree, order, size, diameter, vertex-connectivity and edge-connectivity. Additionally, we show that the girth of L(ΓR) is exactly 3.