Spectra of eccentricity matrix of $H$-join of graphs

Abstract

Let $\varepsilon(G)$ be the eccentricity matrix of a graph $G$ and $Spec(\varepsilon(G))$ be the eccentricity spectrum of $G$. Let $H[G_1,G_2,\ldots, G_k]$ be the $H$-join of graphs $G_1,G_2,\ldots, G_k$ and let $H[G]$ be lexicographic product of $H$ and $G$. This paper finds the eccentricity matrix of a $H$-join of graphs. Using this result, we find (i) $Spec(\varepsilon(H[G]))$ in terms of $Spec(\varepsilon(H))$ if the radius $(rad(H))$ of $H$ is at least three; (ii) $Spec(\varepsilon(K_k[G_1,G_2,\ldots, G_k]))$ if $\Delta(G_i)\leq |V(G_i)|-2$ which generalises some of the results in \cite{Mahato1}; (iii) $Spec(\varepsilon(H[G_1,G_2,\ldots, G_k]))$ if $rad(H)\geq 2$ and $G_i$ is complete whenever $e_H(i)=2$, which generalises some of the results in \cite{Mahato1} and \cite{Wang1}. Finally, we find the characteristic polynomial of $\varepsilon(K_{1,m}[G_0,G_1,\ldots, G_m])$ if $G_i$'s are regular. As a result, we deduce some of the results in \cite{Li}, \cite{Mahato1}, \cite{Patel} and \cite{Wang}.