Abstract
The toughness $t(G)$ of a graph $G=(V,E)$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all $S\subset V$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. We present two tight lower bounds for $t(G)$ in terms of the Laplacian eigenvalues and provide strong support for a conjecture for a better bound which, if true, implies both bounds, and improves and generalizes known bounds by Alon, Brouwer, and the first author. As applications, several new results on perfect matchings, factors and walks from Laplacian eigenvalues are obtained, which leads to a conjecture about Hamiltonicity and Laplacian eigenvalues.
Highlights
Throughout this paper, G = (V, E) is a simple graph of order n with nonempty vertex set V and nonempty edge set E
For a subset S ⊂ V, the subgraph of G induced by V \S is denoted by G − S, and c(G − S) is the number of components of G − S
The toughness t(G) of a graph G is defined as t(G) = min{ c(G|S−|S) }, where the minimum is taken over all proper subsets S ⊂ V such that c(G − S) > 1
Summary
We use λi := λi(G) to denote the i-th largest eigenvalue of the adjacency matrix of G, and let λ = max{|λ2|, |λn|}, that is, λ is the second largest absolute eigenvalue. It was Alon [1] who first showed that for any connected d-regular graph G, t(G) > 13 ( dλd+2λ2 − 1), through which, Alon was able to show that for every t and g there are t-tough graphs of girth strictly greater than g. Almost at the same time, Brouwer [6] independently showed that t(G) > λd −2 for any connected d-regular graph G He conjectured that t(G) λd − 1 in [6,7].
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