Distance matrices on the H-join of graphs: A general result and applications

Abstract

Given a graph $H$ with vertices $1,\ldots ,s$ and a set of pairwise vertex disjoint graphs $G_{1},\ldots ,G_{s},$ the vertex $i$ of $H$ is assigned to $G_{i}.$ Let $G$ be the graph obtained from the graphs $G_{1},\ldots ,G_{s}$ and the edges connecting each vertex of $G_{i}$ with all the vertices of $G_{j}$ for all edge $ij$ of $H.$ The graph $G$ is called the $H-join$ of $G_1,\ldots,G_s$. Let $M(G)$ be a matrix on a graph $G$. A general result on the eigenvalues of $M\left( G\right) $, when the all ones vector is an eigenvector of $M\left( G_{i}\right) $ for $i=1,2,\ldots ,s$, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of $G$ when $G_{1},\ldots ,G_{s}$ are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.