Spectral radius and perfect k-matchings in t-connected graphs

TL;DR

This paper establishes spectral radius conditions that guarantee the existence of perfect k-matchings and fractional perfect matchings in t-connected graphs, advancing spectral graph theory.

Contribution

It provides new tight spectral radius criteria for perfect k-matchings in t-connected graphs, including fractional cases, enhancing understanding of spectral conditions for matchings.

Findings

Spectral radius bounds ensure perfect k-matchings in t-connected graphs.

Tight spectral conditions are identified for fractional perfect matchings.

Results improve previous spectral criteria for graph matchings.

Abstract

A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow\{0,1,2,\ldots,k\}$ with $\sum\limits_{e\in E_G(v)}f(e)\leq k$ for each vertex $v$ of $G$, where $E_G(v)$ is the set of edges incident with $v$ in $G$. A perfect $k$-matching of a graph $G$ is a $k$-matching $f$ satisfying $\sum\limits_{e\in E_G(v)}f(e)=k$ for any vertex $v$ of $G$. A fractional perfect matching of a graph $G$ is a function $f:E(G)\rightarrow [0,1]$ satisfying $\sum\limits_{e\in E_G(v)}f(e)=1$ for any $v\in V(G)$. We denote by $\rho(G)$ the spectral radius of $G$. In this paper, we put forward a tight spectral radius condition for a $t$-connected graph to possess a perfect $k$-matching and a tight spectral radius condition for the existence of a perfect $k$-matching in a $t$-connected graph with a fractional perfect matching.