# The distance spectrum of two new operations of graphs

Document Type: Research Paper

Authors

1 College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China

2 College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China

3 Hunan Normal University

10.22108/toc.2020.116372.1634

Abstract

Let $G$ be a connected graph with vertex set $V(G)={v_1,v_2,cdots,v_n}$. 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 ${mu_1,mu_2,cdots,mu_n}$ 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.

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