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spectra of graphs; squares of graphs; distance regular graphs; association scheme; metrically regular graphs; bipartite graphs; Kneser graph
The present paper deals with the spectra of powers of metrically regular graphs. We prove that there is only two tables of the parameters of an association scheme so that the corresponding metrically regular bipartite graph of diameter $D = 7$ (8 distinct eigenvalues of the adjacency matrix) has the metrically regular square. The results deal with the graphs of the diameter $D < 7$ see [8], [9] and [10].
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[5] Cvetković, D.M., Doob, M., Sachs, H.: Spectra of graphs. Deutscher Verlag der Wissenchaften, Berlin, 1980. MR 0690768
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[7] Smith, J.H.: Some properties of the spectrum of a graph. Comb.Struct. and their Applications, Gordon and Breach, Sci. Publ. Inc., New York-London-Paris, 1970, pp. 403–406. MR 0266799 | Zbl 0249.05136
[8] Vetchý, V.: Metrically regular square of metrically regular bigraphs I. Arch. Math. (Brno) 27b (1991), 183–197. MR 1189214
[9] Vetchý, V.: Metrically regular square of metrically regular bigraphs II. Arch. Math. (Brno) 28 (1992), 17–24. MR 1201862
[10] Vetchý, V.: Metrically regular square of metrically regular bipartite graphs of diameter $D=6$. Arch. Math. (Brno) 29 (1993), 29–38. MR 1242626
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