论文标题
$ c_5/c_6 $的光谱极限上的扩展名给定尺寸
Extensions on spectral extrema of $C_5/C_6$-free graphs with given size
论文作者
论文摘要
令$ \ Mathcal {f} $表示一组图。图$ g $据说为$ \ MATHCAL {F} $ - 如果不包含$ \ Mathcal {f} $的任何元素作为子图。 Turán编号是$ \ Mathcal {f} $中的最大边数 - 带有$ n $ vertices的免费图。众所周知，古典Turán型极端问题旨在研究Turán固定图的数量。在2010年，Nikiforov \ cite {Nik2}类似地提出了Turán类型问题，该问题要求确定$ \ Mathcal {F} $的最大光谱半径 - 带有$ n $ Vertices的免费图。它引起了很多关注，即使经过认真的尝试，许多这样的问题仍然难以捉摸，因此它们被认为是光谱极端图理论中最有趣的问题之一。有趣的是考虑Turán类型问题的另一个频谱，该问题要求确定$ \ Mathcal {f} $的最大光谱半径 - 带有$ M $边缘的免费图。用$ \ Mathcal {g}（m，\ Mathcal {f}）$表示$ \ MATHCAL {F} $的集合 - 免费的图形，带有$ M $边缘的免费图形，没有隔离的顶点。具有最大光谱半径的$ \ Mathcal {G}（M，\ Mathcal {F}）$之间的每个图都称为最大图。令$θ_{p，q，r} $是通过将两个不同的顶点与三个独立的长度$ p，q $和$ r（$）连接起来的theta图（长度指的是边数）。在本文中，我们首先确定$ \ mathcal {g}（m，θ_{1,2,3}）$和$ \ MATHCAL {G}（M，M，θ_{1,2,4}）之间的唯一最大图。然后，我们确定$ \ Mathcal {g}（m，c_5）$（resp。$ \ nathcal {g}（m，c_6）$）之间的所有最大图。这些结果扩展了一些早期的结果。
Let $\mathcal{F}$ denote a set of graphs. A graph $G$ is said to be $\mathcal{F}$-free if it does not contain any element of $\mathcal{F}$ as a subgraph. The Turán number is the maximum possible number of edges in an $\mathcal{F}$-free graph with $n$ vertices. It is well known that classical Turán type extremal problem aims to study the Turán number of fixed graphs. In 2010, Nikiforov \cite{Nik2} proposed analogously a spectral Turán type problem which asks to determine the maximum spectral radius of an $\mathcal{F}$-free graph with $n$ vertices. It attracts much attention and many such problems remained elusive open even after serious attempts, and so they are considered as one of the most intriguing problems in spectral extremal graph theory. It is interesting to consider another spectral Turán type problem which asks to determine the maximum spectral radius of an $\mathcal{F}$-free graph with $m$ edges. Denote by $\mathcal{G}(m,\mathcal{F})$ the set of $\mathcal{F}$-free graphs with $m$ edges having no isolated vertices. Each of the graphs among $\mathcal{G}(m,\mathcal{F})$ having the largest spectral radius is called a maximal graph. Let $θ_{p,q,r}$ be a theta graph formed by connecting two distinct vertices with three independent paths of length $p,q$ and $r,$ respectively (length refers to the number of edges). In this paper, we firstly determine the unique maximal graph among $\mathcal{G}(m,θ_{1,2,3})$ and $\mathcal{G}(m,θ_{1,2,4}),$ respectively. Then we determine all the maximal graphs among $\mathcal{G}(m,C_5)$ (resp. $\mathcal{G}(m,C_6)$) excluding the book graph. These results extend some earlier results.
