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.
Categories: math.CO