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spectra of graphs; square of graphs; bipartite graphs; metrically regular graphs; association scheme
The present paper deals with the spectra of powers of metrically regular graphs. We prove that there is only one table of the parameters of an association scheme so that the corresponding metrically regular bipartite graph of diameter $D = 6$ (7 distinct eigenvalues of the adjacency matrix) has the metrically regular square. The results deal with the graphs of the diameter $D < 6$ see [7] and [8].
[1] Bauer, L.: Association Schemes I. Arch. Math. Brno 17 (1981), 173-184. MR 0672657 | Zbl 0479.05020
[2] Bose, R. C., Shimamoto, T.: Classification and analysis of partially balanced incomplete block design with two association classes. J. Amer. Stat. Assn. 47 (1952), 151-184. MR 0048772
[3] Bose, R. C., Messner, D. M.: On linear associative algebras corresponding to association schemes of partially balanced designs. Ann. Math. Statist. 30 (1959), 21-36. MR 0102157
[4] Cvetković, D. M., Doob, M., Sachs, H.: Spectra of graphs. Deutscher Verlag der Wissenchaften, Berlin, 1980.
[5] Sachs, H.: Über selbstkomplementäre Graphen. Publ. Math. Debrecen 9 (1962), 270-288. MR 0151953 | Zbl 0119.18904
[6] Smith, J. H.: Some properties of the spectrum of a graph. Comb.Struct. and their Applic., Gordon and Breach, Sci. Publ. Inc., New York-London-Paris (1970), 403-406. MR 0266799 | Zbl 0249.05136
[7] Vetchý, V.: Metrically regular square of metrically regular bigraphs I. Arch. Math. Brno 27b (1991), 183-197. MR 1189214
[8] Vetchý, V.: Metrically regular square of metrically regular bigraphs II. Arch. Math. Brno 28 (1992), 17-24. MR 1201862
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