Fractional matching number and spectral radius of nonnegative matrix of graphs

A fractional matching of a graph $G$ is a function $f:E(G)\to [0,1]$ such that for any $v\in V(G)$, ${\sum }_{e\in E_G(v)}f(e)\le 1$ where $E_G(v) = {e \in E(G): e$isincidentwith$v$in$G}$\mathrm{.}Thefractionalmatchingnumberof$G$is$\mu_{f}(G) = \max{\sum_{e\in E(G)} f(e): f$isfractionalmatchingof$G}$\mathrm{.}Foranyrealnumbers$a \ge 0$and$k \in (0, n)$, it is observed that if $n=\mathrm{\mid }V(G)\mathrm{\mid }$ and $\delta (G)>\frac{n-k}{2}$, then ${\mu }_f(G)>\frac{n-k}{2}$. We determine a function $\phi (a,n,\delta ,k)$ and show that for a connected graph $G$ with $n=\mathrm{\mid }V(G)\mathrm{\mid }$, $\delta(G) \leq\frac{n-k}{2}$,spectralradius$\lambda_1(G)$andcomplement$\overline{G}$,eachofthefollowingholds\mathrm{.}(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.