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A t-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most t. Such graphs generalize distance-regular graphs and t-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance-regular graphs. We will generalize Delsarte’s clique bound to 1-walk-regular graphs, Godsil’s multiplicity bound and Terwilliger’s analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show.

We study graphswith spectral radius atmost 3/2√2 and refine results byWoo and Neumaier [R.Woo, A. Neumaier, On graphs whose spectral radius is bounded by 3/2√2, Graphs Combin. 23 (2007) 713–726]. We study the limit points of the spectral radii of certain families of graphs, and apply the results to the problem of minimizing the spectral radius among the graphs with a given number of vertices and diameter. In particular, we consider the cases when the diameter is about half the number of vertices, and when the diameter is near the number of vertices. We prove certain instances of a conjecture posed by Van Dam and Kooij [E.R. Van Dam, R.E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423 (2007) 408–419] and show that the conjecture is false for the other instances.

AMS classifications: 05C50, 05E99

We show that the Hamming graph H(3.q) with diameter three is uniquely determined by its spectrum for q ≥ 36. Moreover, we show that for given integer D ≥ 2, any graph cospectral with the Hamming graph H(D,q) is locally the disjoint union of D copies of the complete graph of size q-1, for q large enough.

2000 Mathematics Subject Classification: 05C50, 05E99

We show that the Hamming graph H(3; q) with diameter three is uniquely determined by its spectrum for q ¸ 36. Moreover, we show that for given integer D ¸ 2, any graph cospectral with the Hamming graph H(D; q) is locally the disjoint union of D copies of the complete graph of size q ¡ 1, for q large enough.

AMS classifications: 05C50; 05E99

AMS classifications: 05E30; 05B20

AMS classification: 05E30

In this paper we investigate the metric hierarchy for distance-regular graphs: in particular, we classify the graphs in the title.