Distance spectral radius and $H_b$-factors in graphs

Let $G$ be a connected graph, and let $b\ge 2$ be an even integer. The distance spectral radius of $G$ is denoted by $\mu (G)$. An $H_b$-factor of $G$ is a spanning subgraph $F$ of $G$ with $d_F(v)\in \{1,3,5,\dots ,b-1,b\}$ for any $v\in V(G)$, where $d_F(v)$ is the degree of $v$ in $F$. Lu and Wang provided a sufficient condition with respect to the number of odd components in $G-S$ for a connected graph $G$ of even order to contain an $H_b$-factor, where $S$ is a vertex subset of $G$ [H. Lu, D. Wang, On Cui-Kano’s characterization problem on graph factors, J. Graph Theory 74 (2013) 335–343]. In this paper, motivated by Lu and Wang’s above result, we establish an upper bound on the distance spectral radius $\mu (G)$ of a connected graph $G$ to guarantee that $G$ contains an $H_b$-factor.