Distance spectrum of the generalized lexicographic product of a graph with a family of regular graphs

In this paper, we determine the distance spectrum of the generalized lexicographic product of a graph G with a family of regular graphs H-x, for x is an element of V(G) by using the adjacency spectrum of H-x is an element of V(G), and the distance spectrum of G. The result is used to construct graphs with only one positive D-eigenvalue and to construct non-isomorphic graphs which have the same distance spectrum. The main theorem of our paper can he viewed as a generalization of a result due to Stevanovic et al. [The distance spectrum and energy of the compositions of regular graphs, Applied Mathematics Letters, 22(2009) 1136-1140], where the authors studied the distance spectrum and energy of the join-based compositions of regular graphs in terms of their adjacency spectrum.