On the Connectivity of Non-Commuting Graph of Finite Rings: Discrete and Computational Geometry, Graphs, and Games - Khuhirun, Borworn; Jantarakhajorn, Khajee; Maneerut, Wanida (Thai Journal of Mathematics, 2023)
The non-commuting graph of a non-commutative ring R, denoted by Γ_R, is a simple graph with vertex set of elements in R except for its center. Two distinct vertices x and y are adjacent if xy eq yx. In this paper, we study the vertex-connectivity and edge-connectivity of a non-commuting graph associated with a finite non-commutative ring R and prove their lower bounds. We show that the edge-connectivity of Γ_R is equal to its minimum degree. The vertex-connectivity and edge-connectivity of Γ_R are determined when R is a non-commutative ring of order p^n where p is a prime number, and n {2,3,4,5}.