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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.)
##### Keywords:
eigenvalues; toughness
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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 [3] Cvetković, D.M.; Doob, M.; Sachs, H., Spectra of graphs, (1979), VEB Berlin, Academic, New York, 1980 [4] Haemers, W.H., Eigenvalue techniques in design and graph theory, (1980), Reidel Dordrecht · Zbl 0429.05013 [5] Haemers, W.H., Interlacing eigenvalues and graphs, Linear algebra appl., 226-228, 593-616, (1995) · Zbl 0831.05044 [6] van den Heuvel, J., Degree and toughness conditions for cycles in graphs, () · Zbl 0840.05046
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