论文标题
图形中的韧性，汉密尔顿性和光谱半径
Toughness, hamiltonicity and spectral radius in graphs
论文作者
论文摘要
在图理论中，对图形中哈密顿周期存在的研究是一个经典问题。通过融合韧性和光谱条件，我们可以从另一个角度考虑Chvátal的猜想：保证在$ t $ tough图中保证哈密顿周期存在的光谱条件是什么？我们首先给出$ 1 $ -tough图的答案，即，如果$ρ（g）\geqρ（m_ {n}）$，则$ g $包含一个哈密顿循环，除非$ g \ g \ cong m_ {n} $，否则$ m_ {n} = n} = k_ {nabla k_ {1} \ nabla k_} $ k_ {n-4}^{+3} $是从$ 3K_ {1} \ cup k_ {n-4} $获得的图表，通过在$ 3K_ {1} $和$ k_ {n-4} $之间添加三个独立边缘。 Brouwer的韧性定理指出，每个$ d $ regented Graph始终具有$ t（g）> \ frac {d}λ-1$，其中$λ$是邻接矩阵的第二大绝对特征值。在本文中，我们根据其光谱半径扩展了结果，即，我们为图形提供了一个频谱条件，以最低度$δ$和$ t $ -tough为1个。
The study of the existence of hamiltonian cycles in a graph is a classic problem in graph theory. By incorporating toughness and spectral conditions, we can consider Chvátal's conjecture from another perspective: what is the spectral condition to guarantee the existence of a hamiltonian cycle among $t$-tough graphs? We first give the answer to $1$-tough graphs, i.e. if $ρ(G)\geqρ(M_{n})$, then $G$ contains a hamiltonian cycle, unless $G\cong M_{n}$, where $M_{n}=K_{1}\nabla K_{n-4}^{+3}$ and $K_{n-4}^{+3}$ is the graph obtained from $3K_{1}\cup K_{n-4}$ by adding three independent edges between $3K_{1}$ and $K_{n-4}$. The Brouwer's toughness theorem states that every $d$-regular connected graph always has $t(G)>\frac{d}λ-1$ where $λ$ is the second largest absolute eigenvalue of the adjacency matrix. In this paper, we extend the result in terms of its spectral radius, i.e. we provide a spectral condition for a graph to be 1-tough with minimum degree $δ$ and to be $t$-tough, respectively.
