ABSTRACT

This paper mainly focuses on spectral conditions for the Hamiltonicity of balanced bipartite graphs and nearly balanced bipartite graphs. Let $\rho \left(G\right)$ and $q\left(G\right)$ be the spectral radius and signless Laplacian spectral radius of a graph G, respectively. One of our main results is the following theorem:

Let G be a balanced bipartite graph of order $2n$ and of minimum degree $\delta \left(G\right)\ge k\ge s\ge 0$.

1. If $n\ge (k+1)(k-s+1)$ and $\rho \left(G\right)\ge \rho \left(B_{n,n}^{n-k,k-s}\right)$, then G is $2s$-hamiltonian unless $G=B_{n,n}^{n-k,k-s}$.

2. If $n\ge (k+1)(k-s+1)$ and $q\left(G\right)\ge q\left(B_{n,n}^{n-k,k-s}\right)$, then G is $2s$-hamiltonian unless $G=B_{n,n}^{n-k,k-s}$.

This theorem generalizes the results in [B. L. Li, B. Ning, Spectral analogues of Erdos' and Moon–Moser's theorems on Hamilton cycles, Linear and Multilinear Algebra, 64 (2016) 2252–2269].

中文翻译：

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二部图的 2s-hamiltonicity 和 2s-traceability 的光谱条件的一些概括

摘要

本文主要关注平衡二部图和近平衡二部图的哈密顿性的谱条件。让$\rho \left(G\right)$和$q\left(G\right)$分别是图G的谱半径和无符号拉普拉斯谱半径。我们的主要结果之一是以下定理：

令G是一个平衡的有序二分图$2n$和最低程度的$\delta \left(G\right)\ge ķ\ge s\ge 0$.

1. 如果$n\ge (ķ+1)(ķ-s+1)$和$\rho \left(G\right)\ge \rho \left({乙}_{n,n}^{n-ķ,ķ-s}\right)$，则G为$2s$- 汉密尔顿除非$G={乙}_{n,n}^{n-ķ,ķ-s}$.

2. 如果$n\ge (ķ+1)(ķ-s+1)$和$q\left(G\right)\ge q\left({乙}_{n,n}^{n-ķ,ķ-s}\right)$，则G为$2s$- 汉密尔顿除非$G={乙}_{n,n}^{n-ķ,ķ-s}$.

该定理概括了 [BL Li, B. Ning, Spectral analogs of Erdos' and Moon-Moser's theorems on Hamilton cycles, Linear and Multilinear Algebra, 64 (2016) 2252–2269] 中的结果。