Perfect matching and distance spectral radius in graphs and bipartite graphs

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.