The distance spectrum of two new operations of graphs

Abstract

Let $G$ be a connected graph with vertex set $V(G)={v_1, v_2,ldots,v_n}$‎. ‎The distance matrix $D=D(G)$ of $G$ is defined so that its $(i,j)$-entry is equal to the distance $d_G(v_i,v_j)$ between the vertices $v_i$ and $v_j$ of $G$‎. ‎The eigenvalues ${mu_1, mu_2,ldots,mu_n}$ of $D(G)$ are the $D$-eigenvalues of $G$ and form the distance spectrum or the $D$-spectrum of $G$‎, ‎denoted by $Spec_D(G)$‎. ‎In this paper‎, ‎we introduce two new operations $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ on graphs $G_1$ and $G_2$‎, ‎and describe the distance spectra of $G_1blacksquare_k G_2$ and $G_1blacklozenge_k G_2$ of regular graphs $G_1$ and $G_2 $ 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.