On graphs with maximum Harary spectral radius

TL;DR

This paper characterizes the graphs that maximize the spectral radius of the Harary matrix within three specific classes of simple connected graphs, focusing on fixed matching numbers and cut edges.

Contribution

It provides a complete characterization of graphs with maximum Harary spectral radius in classes with fixed matching number and cut edges.

Findings

Identifies graphs with maximum Harary spectral radius for fixed matching number.

Determines extremal bipartite graphs with given matching number.

Characterizes graphs with maximum spectral radius given the number of cut edges.

Abstract

Let $G$ be a simple graph with vertex set $V(G) = \{v_1 ,v_2 ,\cdots ,v_n\}$. The Harary matrix $RD(G)$ of $G$, which is initially called the reciprocal distance matrix, is an $n \times n$ matrix whose $(i,j)$-entry is equal to $\frac{1}{d_{ij}}$ if $i\not=j$ and $0$ otherwise, where $d_{ij}$ is the distance of $v_i$ and $v_j$ in $G$. In this paper, we characterize graphs with maximum spectral radius of Harary matrix in three classes of simple connected graphs with $n$ vertices: graphs with fixed matching number, bipartite graphs with fixed matching number, and graphs with given number of cut edges, respectively.