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Some spectral and quasi-spectral characterizations of distance-regular graphs. (English) Zbl 1342.05031
Summary: In this paper we consider the concept of preintersection numbers of a graph. These numbers are determined by the spectrum of the adjacency matrix of the graph, and generalize the intersection numbers of a distance-regular graph. By using the preintersection numbers we give some new spectral and quasi-spectral characterizations of distance-regularity, in particular for graphs with large girth or large odd-girth.

##### MSC:
 05C12 Distance in graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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##### References:
 [1] Biggs, N., Algebraic graph theory, (1974), Cambridge University Press Cambridge, second edition, 1993 · Zbl 0284.05101 [2] Brouwer, A. E.; Haemers, W. H., The gewirtz graph: an exercise in the theory of graph spectra, European J. Combin., 14, 397-407, (1993) · Zbl 0794.05076 [3] Brouwer, A. E.; Haemers, W. H., Spectra of graphs, (2012), Springer, available online at: · Zbl 0794.05076 [4] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin-New York · Zbl 0747.05073 [5] Cámara, M.; Fàbrega, J.; Fiol, M. A.; Garriga, E., Some families of orthogonal polynomials of a discrete variable and their applications to graphs and codes, Electron. J. Combin., 16, 1, (2009) · Zbl 1230.05120 [6] Cvetković, D. M.; Doob, M.; Sachs, H., Spectra of graphs. theory and application, (1982), VEB Deutscher Verlag der Wissenschaften Berlin [7] Dalfó, C.; van Dam, E. R.; Fiol, M. A.; Garriga, E.; Gorissen, B. L., On almost distance-regular graphs, J. Combin. Theory Ser. A, 118, 1094-1113, (2011) · Zbl 1225.05249 [8] van Dam, E. R., Three-class association schemes, J. Algebraic Combin., 10, 69-107, (1999) · Zbl 0929.05096 [9] van Dam, E. R., The spectral excess theorem for distance-regular graphs: a global (over)view, Electron. J. Combin., 15, 1, (2008) · Zbl 1180.05130 [10] van Dam, E. R.; Fiol, M. A., A short proof of the odd-girth theorem, Electron. J. Combin., 19, 3, 12, (2012) [11] van Dam, E. R.; Haemers, W. H., Spectral characterizations of some distance-regular graphs, J. Algebraic Combin., 15, 189-202, (2002) · Zbl 0993.05149 [12] van Dam, E. R.; Haemers, W. H., Which graphs are determined by their spectrum?, Linear Algebra Appl., 373, 241-272, (2003) · Zbl 1026.05079 [13] van Dam, E. R.; Haemers, W. H., Developments on spectral characterizations of graphs, Discrete Math., 309, 576-586, (2009) · Zbl 1205.05156 [14] van Dam, E. R.; Haemers, W. H., An odd characterization of the generalized odd graphs, J. Combin. Theory Ser. B, 101, 486-489, (2011) · Zbl 1234.05157 [15] van Dam, E. R.; Haemers, W. H.; Koolen, J. H.; Spence, E., Characterizing distance-regularity of graphs by the spectrum, J. Combin. Theory Ser. A, 113, 1805-1820, (2006) · Zbl 1105.05076 [16] van Dam, E. R.; Koolen, J. H.; Tanaka, H., Distance-regular graphs, Electron. J. Combin., (2016) · Zbl 1335.05062 [17] Fiol, M. A., Algebraic characterizations of distance-regular graphs, Discrete Math., 246, 111-129, (2002) · Zbl 1025.05060 [18] Fiol, M. A.; Garriga, E., From local adjacency polynomials to locally pseudo-distance-regular graphs, J. Combin. Theory Ser. B, 71, 162-183, (1997) · Zbl 0888.05056 [19] Fiol, M. A.; Gago, S.; Garriga, E., A simple proof of the spectral excess theorem for distance-regular graphs, Linear Algebra Appl., 432, 2418-2422, (2010) · Zbl 1221.05112 [20] Godsil, C. D., Algebraic combinatorics, (1993), Chapman and Hall New York · Zbl 0814.05075 [21] Haemers, W. H., Distance-regularity and the spectrum of graphs, Linear Algebra Appl., 236, 236-278, (1996) · Zbl 0845.05101 [22] Hoffman, A. J., On the polynomial of a graph, Amer. Math. Monthly, 70, 30-36, (1963) · Zbl 0112.14901 [23] Huang, T.; Liu, C., Spectral characterization of some generalized odd graphs, Graphs Combin., 15, 195-209, (1999) · Zbl 0934.05088 [24] Lee, G.-S.; Weng, C.-W., The spectral excess theorem for general graphs, J. Combin. Theory Ser. A, 119, 1427-1431, (2012) · Zbl 1245.05087 [25] Perkel, M., Bounding the valency of polygonal graphs with odd girth, Canad. J. Math., 31, 1307-1321, (1979) · Zbl 0465.20002
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