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Toughness and spectrum of a graph. (English) Zbl 0833.05048
The author proves the following theorem: Let \(G\) be a connected non- complete regular graph of valency \(d\) and let \(\lambda\) be the maximum of the absolute values of the eigenvalues of \(G\) distinct from \(d\). Then the toughness \(t\) of \(G\) satisfies \(t> d/\lambda- 2\).

MSC:
05C35 Extremal problems in graph theory
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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References:
[1] Alon, N., Tough Ramsey graphs without short cycles, (1993), Preprint · Zbl 0826.05039
[2] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer Heidelberg · Zbl 0747.05073
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