On the connectivity of the commuting graph of the matrix ring M_p (GF(q^N ))

Abstract

Let  be a noncommutative ring with identity. The commuting graph associated with , denoted by  is a simple, undirected graph with the vertex set , where two distinct vertices  are adjacent if and only if  In this paper, we study the connectivity and estimate the diameter of the commuting graph associated with the matrix ring of prime degree over the field , where  is a Steinitz number. Specifically, given  as a prime number greater than , we show that the graph  is disconnected if and only if  is finite.