On the distribution of eigenvalues of the reciprocal distance Laplacian matrix of graphs

TL;DR

This paper investigates the eigenvalues of the reciprocal distance Laplacian matrix of graphs, revealing their multiplicities, extremal properties, and spectral determination for several classes of graphs.

Contribution

It characterizes the eigenvalue multiplicities, identifies graphs with extremal spectral radii, and determines the spectrum for key graph families based on the reciprocal distance Laplacian matrix.

Findings

Multiplicity of the eigenvalue n relates to the complement graph's components.

Complete bipartite graphs maximize spectral radius among bipartite graphs.

Star graphs uniquely maximize spectral radius among trees.

Abstract

The reciprocal distance Laplacian matrix of a connected graph $G$ is defined as $RD^L(G)=RT(G)-RD(G)$, where $RT(G)$ is the diagonal matrix of reciprocal distance degrees and $RD(G)$ is the Harary matrix. Since $RD^L(G)$ is a real symmetric matrix, we denote its eigenvalues as $\lambda_1(RD^L(G))\geq \lambda_2(RD^L(G))\geq \dots \geq \lambda_n(RD^L(G))$. The largest eigenvalue $\lambda_1(RD^L(G))$ of $RD^L(G)$ is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of $n$ as a reciprocal distance Laplacian eigenvalue of $RD^L(G)$ is exactly one less than the number of components in the complement graph $\bar{G}$ of $G$. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with $n$ vertices. Also, we show that the star graph $S_n$ is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with $n$ vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph $K_n$, $K_n-e$, the star $S_n$, the complete balanced bipartite graph $K_{\frac{n}{2},\frac{n}{2}}$ and the complete split graph $CS(n,\alpha)$ are all determined from the $RD^L$-spectrum.