Misc,
Fractional matching number and spectral radius of nonnegative matrix of graphs
(2020)cite arxiv:2002.00370.
Abstract
A fractional matching of a graph $G$ is a function $f:E(G) 0,1$ such
that for any $vV(G)$, $\sum_eE_G(v)f(e)1$ where $E_G(v) = \e
E(G): e$ is incident with $v$ in $G\$. The fractional matching number of
$G$ is $\mu_f(G) = \max\\sum_eE(G) f(e): f$ is fractional matching of
$G\$. For any real numbers $a 0$ and $k (0, n)$, it is observed that
if $n = |V(G)|$ and $\delta(G) > n-k2$, then
$\mu_f(G)>n-k2$. We determine a function $\varphi(a, n,\delta, k)$
and show that for a connected graph $G$ with $n = |V(G)|$, $\delta(G)
łeqn-k2$, spectral radius $łambda_1(G)$ and complement
$G$, each of the following holds. (i) If
$łambda_1(aD(G)+A(G))<\varphi(a, n, \delta, k),$ then
$\mu_f(G)>n-k2.$ (ii) If
$łambda_1(aD(G)+A(G))<(a+1)(\delta+k-1),$ then
$\mu_f(G)>n-k2.$ 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.
