论文标题
具有给定尺寸和奇数的图形的光谱半径
Spectral radius of graphs with given size and odd girth
论文作者
论文摘要
令$ \ Mathcal {g}（m，k）$为尺寸$ m $和奇数围绕最短周期的长度）$ k $的图表。在本文中，我们确定$ \ mathcal {g}（m，k）$之间的光谱半径最大化的图表。作为副产品，我们表明，有一个数字$η（m）> \ sqrt {m-k+3} $，使每个非双方图$ g $带有尺寸$ $ m $和spectral radius $ρ\geη（m）$必须包含的长度小于$ k $ g $ m $ grouns和$ g \ g cong \ g cong \ g cong \ ge cong \ ge \ ge \细分边缘$ k-2 $ times完整的两分$ k_ {2，\ frac {m-k+2} {2}}} $。该结果意味着[离散数学的主要结果。 345（2022）]和\ cite {li-peng}，并在\ cite {li-peng}中解决了猜想。
Let $\mathcal{G}(m,k)$ be the set of graphs with size $m$ and odd girth (the length of shortest odd cycle) $k$. In this paper, we determine the graph maximizing the spectral radius among $\mathcal{G}(m,k)$ when $m$ is odd. As byproducts, we show that, there is a number $η(m)>\sqrt{m-k+3}$ such that every non-bipartite graph $G$ with size $m$ and spectral radius $ρ\ge η(m)$ must contains an odd cycle of length less than $k$ unless $m$ is odd and $G\cong SK_{k,m}$, which is the graph obtained by subdividing an edge $k-2$ times of complete bipartite $K_{2,\frac{m-k+2}{2}}$. This result implies the main results of [Discrete Math. 345 (2022)] and \cite{li-peng}, and settles the conjecture in \cite{li-peng} as well.
