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On almost distance-regular graphs. (English) Zbl 1225.05249
Summary: Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study ‘almost distance-regular graphs’. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant.
One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called $$m$$-walk-regularity. Another studied concept is that of $$m$$-partial distance-regularity or, informally, distance-regularity up to distance $$m$$. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of $$(\ell ,m)$$-walk-regularity.
We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs
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##### References:
 [1] Beezer, R.A., Distance polynomial graphs, (), 51-73 [2] Biggs, N., Algebraic graph theory, (1994), Cambridge University Press Cambridge, second ed., 1973 · Zbl 0797.05032 [3] Bloom, G.S.; Quintas, L.W.; Kennedy, J.W., Distance degree regular graphs, (), 95-108 · Zbl 0476.05070 [4] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin/New York · Zbl 0747.05073 [5] Dalfó, C.; Fiol, M.A.; Garriga, E., On k-walk-regular graphs, Electron. J. combin., 16, 1, (2009), #R47 · Zbl 1226.05107 [6] Dalfó, C.; Fiol, M.A.; Garriga, E., Characterizing $$(\ell, m)$$-walk-regular graphs, Linear algebra appl., 433, 1821-1826, (2010) · Zbl 1213.05164 [7] 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 [8] van Dam, E.R., The spectral excess theorem for distance-regular graphs: a global (over)view, Electron. J. combin., 15, 1, (2008), #R129 · Zbl 1180.05130 [9] Fiol, M.A., Algebraic characterizations of distance-regular graphs, Discrete math., 246, 111-129, (2002) · Zbl 1025.05060 [10] 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 [11] Fiol, M.A.; Garriga, E., The alternating and adjacency polynomials, and their relation with the spectra and diameters of graphs, Discrete appl. math., 87, 77-97, (1998) · Zbl 0914.05050 [12] 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 [13] Fiol, M.A.; Garriga, E., On the algebraic theory of pseudo-distance-regularity around a set, Linear algebra appl., 298, 115-141, (1999) · Zbl 0984.05060 [14] 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 [15] Fiol, M.A.; Garriga, E.; Yebra, J.L.A., Boundary graphs: the limit case of a spectral property, Discrete math., 226, 155-173, (2001) · Zbl 0965.05068 [16] Godsil, C.D., Algebraic combinatorics, (1993), Chapman and Hall New York · Zbl 0814.05075 [17] 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 [18] Hilano, T.; Nomura, K., Distance degree regular graphs, J. combin. theory ser. B, 37, 96-100, (1984) · Zbl 0567.05028 [19] Hoffman, A.J., On the polynomial of a graph, Amer. math. monthly, 70, 30-36, (1963) · Zbl 0112.14901 [20] Huang, T.; Huang, Y.; Liu, S.-C.; Weng, C., Partially distance-regular graphs and partially walk-regular graphs, (2007) [21] Klin, M.; Muzychuk, M.; Ziv-Av, M., Higmanian rank-5 association schemes on 40 points, Michigan math. J., 58, 255-284, (2009) · Zbl 1284.05339 [22] Martin, W.J.; Tanaka, H., Commutative association schemes, European J. combin., 30, 1497-1525, (2009) · Zbl 1228.05317 [23] McKay, B.D.; Stanton, R.G., The current status of the generalised Moore graph problem, (), 21-31 · Zbl 0422.05044 [24] Powers, D.L., Partially distance-regular graphs, (), 991-1000 · Zbl 0841.05098 [25] Rowlinson, P., Linear algebra, (), 86-99 · Zbl 0878.05059 [26] Royle, G.; Conder, M.; McKay, B.; Dobscanyi, P., Cubic symmetric graphs, (July 2009), (The Foster Census) [27] Sampels, M., Vertex-symmetric generalized Moore graphs, Discrete appl. math., 138, 195-202, (2004) · Zbl 1034.05019 [28] Weichsel, P.M., On distance-regularity in graphs, J. combin. theory ser. B, 32, 156-161, (1982) · Zbl 0477.05047
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