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arXiv · 2310.09523

Toughness and spectral radius in graphs

Yuanyuan Chen·Dandan Fan·Huiqiu Lin
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Abstract

The Brouwer's toughness conjecture states that every $d$-regular connected graph always has $t(G)>\frac{d}{\lambda}-1$ where $\lambda$ is the second largest absolute eigenvalue of the adjacency matrix. In 1988, Enomoto introduced a variation of toughness $\tau(G)$ of a graph $G$. By incorporating the variation of toughness and spectral conditions, we provide spectral conditions for a graph to be $\tau$-tough ($\tau\geq 2$ is an integer) and to be $\tau$-tough ($\frac{1}{\tau}$ is a positive integer) with minimum degree $\delta$, respectively. Additionally, we also investigate a analogous problem concerning balanced bipartite graphs.

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Yuanyuan Chen, Dandan Fan, Huiqiu Lin. 2023-10-14. Toughness and spectral radius in graphs. https://arxiv.org/abs/2310.09523

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