论文标题
图形非负矩阵的分数匹配数和光谱半径
Fractional matching number and spectral radius of nonnegative matrix of graphs
论文作者
论文摘要
图$ g $的分数匹配是一个函数$ f：e（g）\ to [0,1] $，因此对于v（g）$中的任何$ v \，$ \ sum_ {e _g（v）} f（e）f（e）\ leq 1 $ in $ e_g（v）$ e_g（v）= \ e \ in e e（g）$ $ $ $ $ $ g $的分数匹配数为$μ_{f}（g）= \ max \ {\ sum_ {e \ in E（g）} f（e）：f $是$ g \ \} $的分数匹配。对于任何实际数字$ a \ ge 0 $和$ k \ in（0，n）$，可以观察到，如果$ n = | v（g）| $和$δ（g）> \ frac {n-k} {2} {2} $，则$μ_________________________________________________{f}（g）（g）> \ frac> \ frac {n-k} {n-k} $。 We determine a function $φ(a, n,δ, k)$ and show that for a connected graph $G$ with $n = |V(G)|$, $δ(G) \leq\frac{n-k}{2}$, spectral radius $λ_1(G)$ and complement $\overline{G}$, each of the following holds. （i）如果$λ_{1}（ad（g）+a（g））<φ（a，n，δ，k），然后$μ__{f}（g）（g）> \ frac {n-k} {2} {2} {2} {$ $ （ii）如果$λ_{1}（AD（\+edline {g}）+a（\ overline {g}）））<（a+1）（δ+k-1），$ then $μ__{f}（g）> \ frac> \ frac {n-k} {n-k} {2} {2} {2} {2}。$ $ 作为推论，获得了足够的分数完美匹配的光谱条件以及涉及$ q $ index和$a_α$ spectral半径的类似结果，并且以前的光谱在[欧洲J. Combin。 55（2016）144-148]扩展。
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 $μ_{f}(G) = \max\{\sum_{e\in E(G)} f(e): f$ 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 $δ(G) > \frac{n-k}{2}$, then $μ_{f}(G)>\frac{n-k}{2}$. We determine a function $φ(a, n,δ, k)$ and show that for a connected graph $G$ with $n = |V(G)|$, $δ(G) \leq\frac{n-k}{2}$, spectral radius $λ_1(G)$ and complement $\overline{G}$, each of the following holds. (i) If $λ_{1}(aD(G)+A(G))<φ(a, n, δ, k),$ then $μ_{f}(G)>\frac{n-k}{2}.$ (ii) If $λ_{1}(aD(\overline{G})+A(\overline{G}))<(a+1)(δ+k-1),$ then $μ_{f}(G)>\frac{n-k}{2}.$ As corollaries, sufficient spectral condition for fractional perfect matchings and analogous results involving $Q$-index and $A_α$-spectral radius are obtained, and former spectral results in [European J. Combin. 55 (2016) 144-148] are extended.
