Fractional matching number and spectral radius of nonnegative matrix of graphs

Abstract

A fractional matching of a graph $G$ is a function $f:E(G) \to [0,1]$ such that for any $v\in V(G)$, $\sum_{e\in E_G(v)}f(e)\leq 1$ where $E_G(v) = \{e \in E(G): e$ is incident with $v$ in $G\}$. The fractional matching number of $G$ is $\mu_{f}(G) = \max\{\sum_{e\in E(G)} f(e): f$ is fractional matching of $G\}$. For any real numbers $a \ge 0$ and $k \in (0, n)$, it is observed that if $n = |V(G)|$ and $\delta(G) > \frac{n-k}{2}$, then $\mu_{f}(G)>\frac{n-k}{2}$. We determine a function $\varphi(a, n,\delta, k)$ and show that for a connected graph $G$ with $n = |V(G)|$, $\delta(G) \leq\frac{n-k}{2}$, spectral radius $\lambda_1(G)$ and complement $\overline{G}$, each of the following holds. (i) If $\lambda_{1}(aD(G)+A(G))<\varphi(a, n, \delta, k),$ then $\mu_{f}(G)>\frac{n-k}{2}.$ (ii) If $\lambda_{1}(aD(\overline{G})+A(\overline{G}))<(a+1)(\delta+k-1),$ then $\mu_{f}(G)>\frac{n-k}{2}.$ As corollaries, sufficient spectral condition for fractional perfect matchings and analogous results involving $Q$-index and $A_{\alpha}$-spectral radius are obtained, and former spectral results in [European J. Combin. 55 (2016) 144-148] are extended.