Distance spectral radius and perfect matchings in graphs with given fractional property

TL;DR

This paper investigates the relationship between the distance spectral radius and the existence of perfect matchings in k-connected graphs with fractional perfect matchings, establishing spectral conditions for perfect matchings.

Contribution

It provides a spectral bound involving the distance spectral radius that guarantees the presence of a perfect matching in such graphs, characterizing extremal cases.

Findings

If the distance spectral radius is below a certain bound, the graph has a perfect matching.

The extremal graph achieving equality is explicitly characterized.

The result applies to k-connected graphs of even order with fractional perfect matchings.

Abstract

A matching in a graph $G$ is a set of independent edges in $G$. A perfect matching in a graph $G$ is a matching which saturates all the vertices of $G$. A fractional perfect matching in a graph $G$ is a function $h:E(G)\rightarrow [0,1]$ such that $\sum\limits_{e\in E_G(v)}h(e)=1$ for every $v\in V(G)$, where $E_G(v)$ is the set of edges incident to $v$ in $G$. Clearly, the existence of a fractional perfect matching in a graph is a necessary condition for the graph to possess a perfect matching. Let $G$ be a $k$-connected graph of even order $n$ with a fractional perfect matching, where $k$ is a positive integer. We denote by $\mu(G)$ the distance spectral radius of $G$. In this paper, we prove that if $n\geq8k+6$ and $\mu(G)\leq\mu(K_k\vee(kK_1\cup K_3\cup K_{n-2k-3}))$, then $G$ contains a perfect matching unless $G=K_k\vee(kK_1\cup K_3\cup K_{n-2k-3})$.