On Distance-Regular Graphs with Smallest Eigenvalue at Least −m-m

A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer $m\geq 2$, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least $-m$, diameter at least three and intersection number $c_2 \geq 2$

Spectral Characterization of the Hamming Graphs

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.Hamming graphs;distance-regular graphs;eigenvalues of graphs

On geometric distance-regular graphs with diameter three

In this paper we study distance-regular graphs with intersection array {(t + 1)s. ts. (t - 1)(s + 1 - psi); 1, 2, (t + 1)psi} (1) where s. t. psi are integers satisfying t >= 2 and 1 = 2, there are only finitely many distance-regular graphs of order (s, t) with mallest eigenvalue -t -1, diameter D = 3 and intersection number c(2) = 2 except for Hamming graphs with diameter three. Moreover, we will show that if a distance-regular graph with intersection array (1) for t = 2 exists then (s, psi) = (15, 9). As Gavrilyuk and Makhnev (2013)[9] proved that the case (s, psi) = (15, 9) does not exist, this enables us to finish the classification of geometric distance-regular graphs with smallest eigenvalue -3, diameter D >= 3 and c(2) >= 2 which was started by the first author (Bang, 2013)[1]. (C) 2013 Elsevier Ltd. All rights reserved.X1121Ysciescopu