On the generalized distance spectrum of some connected graphs of diameter at most two

Abstract

Let G be a simple, undirected connected graph. The generalized distance matrix of a graph G, denoted by D_\alpha(G) is the convex combination of the distance matrix D(G) and the diagonal matrix of the vertex transmissions Tr(G). It is of the form D_\alpha(G) = \alpha Tr(G) + (1-\alpha)D(G). In this paper, we determine the spectrum of the generalized distance matrix of the join of regular graphs of diameter at most two, viz. the weakly zero divisor graph and the zero divisor graph on the ring Z_n, for some values of n. For a commutative ring R with unity, the zero-divisor graph, denoted by \Gamma(R), is an undirected simple graph whose vertex set is the set of all non zero and zero-divisors of R and two distinct vertices x and y are adjacent if and only if xy =0. The weakly zero-divisor graph of R denoted by W \Gamma (R) is the simple undirected graph whose vertices are the non zero zero-divisors of R and two distinct vertices x and y are adjacent if and only if there exist r in ann(x) and s in ann(y) such that rs = 0.