The distance spectrum of two new operations of graphs

Let $G$ be a connected graph with vertex set $V(G)=\{v_۱, v_۲,\ldots,v_n\}$‎. ‎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 ${\mu_۱, \mu_۲,\ldots,\mu_n}$ 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 $G_۱\blacksquare_k G_۲$ and $G_۱\blacklozenge_k G_۲$ on graphs $G_۱$ and $G_۲$‎, ‎and describe the distance spectra of $G_۱\blacksquare_k G_۲$ and $G_۱\blacklozenge_k G_۲$ of regular graphs $G_۱$ and $G_۲ $ 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.