arXiv 2207.030452022-07math.CO

具有禁止子图的给定大小图的最大谱半径

The maximum spectral radius of graphs of given size with forbidden subgraph

Xiaona Fang, Lihua You

备注：: 15 pages, 3 figures

摘要：

设$G$是一个图的大小为$m$，$ρ(G)$是它的邻接矩阵的谱半径。如果一个图不包含与$F$同构的子图，则称该图是$F$ -free的.本文证明了：如果$G$是一个无$K_{2,r+1}$的非星图，且$m\geq (4r+2)^2+1$，则$ρ(G)\leq ρ(S_m^1)$，等式当且仅当$G\cong S_m^1$ .最近，Li，Sun和Wei证明了：对于任意的$θ_{1,2,3}$ -free图，若图的大小为$m\geq 8$，$ρ(G)\leq \frac{1+\sqrt{4m-3}}{2}$，则等式当且仅当图的大小为$G\cong S_{\frac{m+3}{2},2}$ .然而，当$m$是偶数时，这个界限是不可达到的。我们证明了如果$G$是$θ_{1,2,3}$ -free的，而$G\ncong S_{\frac{m+3}{2},2}$有$m\geq 22$，那么当$m$是偶数时，则$ρ(G)\leq ρ(F_{m,1})$相等当且仅当$G\cong F_{m,1}$，当$m$是奇数时，则$ρ(G)\leq ρ(F_{m,2})$相等当且仅当$G\cong F_{m,2}$。

英文摘要：

Let $G$ be a graph of size $m$ and $ρ(G)$ be the spectral radius of its adjacency matrix. A graph is said to be $F$-free if it does not contain a subgraph isomorphic to $F$. In this paper, we prove that if $G$ is a $K_{2,r+1}$-free non-star graph with $m\geq (4r+2)^2+1$, then $ρ(G)\leq ρ(S_m^1)$, with equality if and only if $G\cong S_m^1$. Recently, Li, Sun and Wei showed that for any $θ_{1,2,3}$-free graph of size $m\geq 8$, $ρ(G)\leq \frac{1+\sqrt{4m-3}}{2}$, with equality if and only if $G\cong S_{\frac{m+3}{2},2}$. However, this bound is not attainable when $m$ is even. We proved that if $G$ is $θ_{1,2,3}$-free and $G\ncong S_{\frac{m+3}{2},2}$ with $m\geq 22$, then $ρ(G)\leq ρ(F_{m,1})$ if $m$ is even, with equality if and only if $G\cong F_{m,1}$, and $ρ(G)\leq ρ(F_{m,2})$ if $m$ is odd, with equality if and only if $G\cong F_{m,2}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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