On pseudo-distance-regularity. (English) Zbl 0978.05050

The author uses (cosines of the) angles between eigenspaces of (the adjacency matrix of) a graph \(G\) on \(n\) vertices and the \(i\)th axis of a standard basis of \(\mathbb{R}^n\) (cf., e.g., the book by the reviewer, P. Rowlinson and S. Simić [Eigenspaces of graphs (Encyclopedia of Mathematics and Its Applications 66. Cambridge: Cambridge University Press) (1997; Zbl 0878.05057)]) and calls them the \(i\)-local multiplicities of the eigenvalues of \(G\). On this basis, in a previous paper by the author, E. Garriga and J. L. A. Yebra [J. Comb. Theory, Ser. B 68, No. 2, 179-205 (1996; Zbl 0861.05064)] the concept of local pseudo-distance regularity of a graph has been introduced. In the paper under review the author studies some properties of locally distance-regular graphs.


05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text: DOI


[1] N. Biggs, Algebraic Graph Theory, Cambridge University Press, Cambridge, MA, 1974; second ed., 1993 · Zbl 0284.05101
[2] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer Berlin · Zbl 0747.05073
[3] Cvetković, D.; Doob, M., Developments in the theory of graph spectra, Linear and multilinear algebra, 18, 153-181, (1985) · Zbl 0615.05039
[4] Cvetković, D.; Rowlinson, P.; Simić, S., Eigenspaces of graphs, (1997), Cambridge University Press Cambridge, MA · Zbl 0878.05057
[5] Fiol, M.A., Eigenvalue interlacing and weight parameters of graphs, Linear algebra appl., 290, 1-3, 275-301, (1999) · Zbl 0933.05099
[6] Fiol, M.A.; Garriga, E., From local adjacency polynomials to locally pseudo-distance-regular graphs, J. combin. theory ser. B, 71, 2, 162-183, (1997) · Zbl 0888.05056
[7] Fiol, M.A.; Garriga, E., On the algebraic theory of pseudo-distance-regularity around a set, Linear algebra appl., 298, 1-3, 115-141, (1999) · Zbl 0984.05060
[8] Fiol, M.A.; Garriga, E.; Yebra, J.L.A., Locally pseudo-distance-regular graphs, J. combin. theory ser. B, 68, 179-205, (1996) · Zbl 0861.05064
[9] M.A. Fiol, E. Garriga, J.L.A.Yebra, Boundary graphs: the limit case of a spectral property, Discrete Mathematics, to appear · Zbl 0965.05068
[10] E. Garriga, Contribució a la Teoria Espectral de Grafs: Problemes Mètrics i Distancia-Regularitat, (Catalan) Ph.D. Thesis, Universitat Politècnica de Catalunya, 1997
[11] C.D. Godsil, Graphs, groups, and polytopes, in: D.A. Holton, J. Seberry (Eds.), Combinatorial Mathematics, Lecture Notes in Mathematics, vol. 686, Springer, Berlin, 1978, pp. 157-164
[12] Godsil, C.D., Bounding the diameter of distance regular graphs, Combinatorica, 8, 4, 333-343, (1988) · Zbl 0657.05047
[13] C.D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993
[14] Godsil, C.D.; McKay, B.D., Feasibility conditions for the existence of walk-regular graphs, Linear algebra appl., 30, 51-61, (1980) · Zbl 0452.05045
[15] P. Rowlinson, Linear algebra, in: Graph Connections, Oxford University Press, New York, 1997, pp. 86-99 · Zbl 0878.05059
[16] G. Szegö, Orthogonal Polynomials, fourth ed., AMS, Providence, RI, 1975
[17] Terwilliger, P., A characterization of P- and Q-polynomial association schemes, J. combin. theory ser. A, 45, 1, 8-26, (1987) · Zbl 0663.05016
[18] Terwilliger, P., The subconstituent algebra of an association scheme, part I, J. algebraic. combin., 1, 363-388, (1992) · Zbl 0785.05089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.