University of IsfahanTransactions on Combinatorics2251-86579320200901The distance spectrum of two new operations of graphs1251382446710.22108/toc.2020.116372.1634ENZikaiTangKey Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta-
tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaRenfangWuKey Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta-
tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaHanlinChenKey Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta-
tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. ChinaHanyuanDengKey Laboratory of Computing and Stochastic Mathematics (Ministry of Education), College of Mathematics and Sta-
tistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China0000-0003-1680-2473Journal Article20190409Let $G$ 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 $G_1\blacksquare_k G_2$ and $G_1\blacklozenge_k G_2$ on graphs $G_1$ and $G_2$, and describe the distance spectra of $G_1\blacksquare_k G_2$ and $G_1\blacklozenge_k G_2$ 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.https://toc.ui.ac.ir/article_24467_d2bacaa1da976e615698f1efa93b6605.pdf