Perfect matching and distance spectral radius in graphs and bipartite graphs

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 $\hat{I}{»}_1(D(G))$ be the distance spectral radius of $G$. Then the following statements are true. \noindent$\mathrm{I})$ If $4\le n\le 10$ and ${\hat{I}»}_1(D\left(G\right))\le {\hat{I}»}_1(D(S_{n,\frac{n}{2}-1}))$, 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$\mathrm{II})$ If $n\ge 12$ and ${\hat{I}»}_1(D\left(G\right))\le {\hat{I}»}_1(D(G^{\ast }))$, then $G$ contains a perfect matching unless $G\cong G^{\ast }$ where $G^{\ast }\cong K_1\vee (K_{n-3}\cup 2K_1)$. Moreover, if $G$ is a connected $2n$-vertex balanced bipartite graph with $\hat{I}{»}_1(D(G))\le \hat{I}{»}_1(D(B_{n-1,n-2}))$, 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.