Perfect matching and distance spectral radius in graphs and bipartite graphs

TL;DR

This paper establishes spectral radius conditions based on the distance matrix to guarantee the existence of perfect matchings in graphs and bipartite graphs, extending spectral graph theory results.

Contribution

It introduces new spectral radius bounds involving the distance matrix that ensure perfect matchings in graphs and bipartite graphs, identifying extremal structures.

Findings

Spectral radius bounds guarantee perfect matchings for certain graph classes.

Characterization of extremal graphs that meet the spectral bounds without having perfect matchings.

Extension of spectral conditions from eigenvalues to the distance spectral radius in graph theory.

Abstract

A perfect matching in a graph $G$ is a set of nonadjacent edges covering every vertex of $G$. Motivated by recent progress on the relations between the eigenvalues and the matching number of a graph, in this paper, we aim to present a distance spectral radius condition to guarantee the existence of a perfect matching. Let $G$ be an $n$-vertex connected graph where $n$ is even and $\lambda_{1}(D(G))$ be the distance spectral radius of $G$. Then the following statements are true. \noindent$\rm{I)}$ If $4\le n\le10$ and ${\lambda }_{1} (D\left(G\right))\le {\lambda }_{1} (D(S_{n,{\frac{n}{2}}-1}))$, then $G$ contains a perfect matching unless $G\cong S_{n,{\frac{n}{2}-1}}$ where $S_{n,{\frac{n}{2}-1}}\cong K_{{\frac{n}{2}-1}}\vee ({\frac{n}{2}+1})K_1$. \noindent$\rm{II)}$ If $n\ge 12$ and ${\lambda }_{1} (D\left(G\right))\le {\lambda }_{1} (D(G^*))$, then $G$ contains a perfect matching unless $G\cong G^*$ where $G^*\cong K_1\vee (K_{n-3}\cup2K_1)$. Moreover, if $G$ is a connected $2n$-vertex balanced bipartite graph with $\lambda_{1}(D(G))\le \lambda_{1}(D(B_{n-1,n-2})) $, then $G$ contains a perfect matching, unless $G\cong B_{n-1,n-2}$ where $B_{n-1,n-2}$ is obtained from $K_{n,n-2}$ by attaching two pendent vertices to a vertex in the $n$-vertex part.