arXiv 2002.003702020-02-02math.CO

图的非负矩阵的分数匹配数与谱半径

Fractional matching number and spectral radius of nonnegative matrix of graphs

Ruifang Liu, Hong-Jian Lai, Litao Guo, Jie Xue

摘要：

图$G$的分数匹配是函数$f:E(G) \to [0,1]$，使得对于任何$v\in V(G)$，$\sum_{e\in E_G(v)}f(e)\leq 1$，其中$E_G(v) = \{e \in E(G): e$与$G\}$中的$v$关联。$G$的分数匹配数是$μ_{f}(G) = \max\{\sum_{e\in E(G)} f(e): f$是$G\}$的分数匹配。对于任何实数$a \ge 0$和$k \in (0, n)$，观察到如果$n = |V(G)|$和$δ(G) > \frac{n-k}{2}$，则$μ_{f}(G)>\frac{n-k}{2}$。我们确定了一个函数$φ(a, n,δ, k)$，并证明了对于具有$n = |V(G)|$，$δ(G) \leq\frac{n-k}{2}$，谱半径$λ_1(G)$和补$\overline{G}$的连通图$G$，以下每一个都成立。(i)如果$λ_{1}(aD(G)+A(G))<φ(a, n, δ, k),$则$μ_{f}(G)>\frac{n-k}{2}.$（ii）如果$λ_{1}(aD(\overline{G})+A(\overline{G}))<(a+1)(δ+k-1),$则$μ_{f}(G)>\frac{n-k}{2}.$作为推论，得到了分数完美匹配的充分谱条件以及涉及$Q$ -指标和$A_α$ -谱半径的类似结果，推广了[European J. Combin. 55（2016）144-148]延长。

英文摘要：

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.

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摘要：

英文摘要：

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