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

S. Balamoorthy, T. Kavaskar

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 $Δ(G_i)\leq |V(G_i)|-2$ 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}.

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

Abstract

Let be the eccentricity matrix of a graph and be the eccentricity spectrum of . Let be the -join of graphs and let be lexicographic product of and . This paper finds the eccentricity matrix of a -join of graphs. Using this result, we find (i) in terms of if the radius of is at least three; (ii) if which generalises some of the results in \cite{Mahato1}; (iii) if and is complete whenever , which generalises some of the results in \cite{Mahato1} and \cite{Wang1}. Finally, we find the characteristic polynomial of if 's are regular. As a result, we deduce some of the results in \cite{Li}, \cite{Mahato1}, \cite{Patel} and \cite{Wang}.

Paper Structure

This paper contains 4 sections, 35 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Eccentricity matrix of $H$-join of graphs
  3. Spectrum of $H$-join of some graphs
  4. Spectrum of $K_{1,m}$-join of regular graphs

Key Result

Lemma 1

Let $A, B, C$ and $D$ be matrices such that $M= $. If $A$ is invertible, then $\det(M) = \det(A) \det(D - CA^{-1}B).$

Theorems & Definitions (58)

  • Lemma 1: Cvet
  • Lemma 2: Cvetkovic
  • Lemma 3: Cvetkovic
  • Lemma 4: Roger
  • Lemma 5: Roger1
  • Theorem 1: Saravanan
  • Lemma 6: Saravanan
  • Lemma 7: Wang1
  • Theorem 2: You
  • Lemma 8
  • ...and 48 more