论文标题
给定尺寸的图形的最大光谱半径和禁止子图
The maximum spectral radius of graphs of given size with forbidden subgraph
论文作者
论文摘要
令$ g $为$ m $和$ρ（g）$的图表是其邻接矩阵的光谱半径。如果图形不包含$ f $ $ f $的子图，则据说该图是$ f $ f的。在本文中，我们证明，如果$ g $是$ k_ {2，r+1} $ - 带有$ m \ geq（4r+2）^2+1 $的免费非星形图，则是$ρ（g）\ leq p（s_m^1）$，如果是均等的，则只有$ g \ g \ g \ cong \ can_m^1 $。最近，Li，Sun和Wei表明，对于任何$θ_{1,2,3} $ - 大小的免费图$ M \ geq 8 $，$ρ（g）\ leq \ frac \ frac {1+ \ sqrt {4m-3}}} s _ {\ frac {m+3} {2}，2} $。但是，当$ m $均匀时，这是无法实现的。我们证明，如果$ g $是$θ_{1,2,3} $ - 免费和$ g \ ncong s _ {\ frac {\ frac {m+3} {2} {2} {2} $，带有$ m \ geq 22 $，然后是$ρ（g）\ leq p（m，1}） f_ {m，1} $和$ρ（g）\ leqρ（f_ {m，2}）$如果$ m $是奇怪的，则在且仅当$ g \ cong f_ {m，2} $时具有平等性。
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 $G\cong F_{m,2}$.
