Matching, odd $[1,b]$-factor and distance spectral radius of graphs with given some parameters

Abstract: Let $\mu(G)$ denote the the distance spectral radius of $G$. A matching in a graph $G$ is a set of disjoint edges of $G$. The maximum size of a matching in $G$ is called the matching number of $G$, denoted by $\alpha(G)$. In this paper, we give a sharp upper bound in terms of the distance spectral radius to guarantee $\alpha(G)>\frac{n-k}{2}$ in a graph $G$ with given connectivity. Let $a$ and $b$ be two positive integers with $a\le b$. An $[a, b]$-factor of a graph $G$ is a spanning subgraph $G_0$ such that for every vertex $v\in V (G)$, the degree $d_{G_0}(v)$ of $v$ in $H$ satisfies $a\le d_{G_0}(v)\le b$. In this paper, we present a sharp upper bound in terms of distance spectral radius for the existence of an odd $[1,b]$-factor in a graph with given minimum degree $\delta$.