the distance spectrum of two new operations of graphs

Let g be a connected graph with vertex set v (g) = {v1, v2, . . . , vn}. the distance matrix d = d(g) of g is defined so that its (i, j)-entry is equal to the distance dg(vi, vj ) between the vertices vi and vj of g. the eigenvalues µ1, µ2, . . . , µn of d(g) are the d-eigenvalues of g and form the distance spectrum or the d-spectrum of g, denoted by specd(g). in this paper, we introduce two new operations g1■kg2 and g1♦kg2 on graphs g1 and g2, and describe the distance spectra of g1■kg2 and g1♦kg2 of regular graphs g1 and g2 in terms of their adjacency spectra. by using these results, we obtain some new integral adjacency spectrum graphs, integral distance spectrum graphs and a number of families of sets of noncospectral graphs with equal distance energy.