Distance-Regular Graphs

Abstract

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.

A spectral characterization of the s-clique extension of the square grid graphs

“…For undefined terminologies, see the next section. For the definitions related to distance-regular graphs, see [1,6].…”

Non-Bipartite Distance-Regular Graphs with a Small Smallest Eigenvalue

“…Then as m ≥ k ≥ 71, by Equation (12), we obtain c 3 ≤ 0.2166k. It implies |u 4 | ≥ 0.3344 by Equation (7). Consider the intersection matrix L, and we see that k 2 + θ 2 ≤ tr(L 2 ) ≤ k 2 + 6k + 4c 5 k − c 2 5 , which implies c 5 ≥ 0.1440k.…”

“…For more background, see [4] and [7]. All the graphs considered in this paper are finite, undirected and simple.…”

“…We denote m i the multiplicity of θ i . Note that the D + 1 distinct eigenvalues of Γ are precisely the eigenvalues of L (see [7,Proposition 2.7]).…”

On some distance-regular graphs with many vertices

“…We assume that the reader is familiar with the basic facts of the group theory, the theory of strongly regular graphs and the theory of distance-regular graphs. We refer the reader to [8,20] for relevant background reading in the group theory, to [3,23] for the theory of strongly regular graphs, and to [5,12] for the theory of distance-regular graphs.…”