A recursive construction of the regular exceptional graphs with least eigenvalue –2

Abstract

In spectral graph theory a graph with least eigenvalue −2 is exceptional if it is connected, has least eigenvalue greater than or equal to −2, and it is not a generalized line graph. A (κ, τ)-regular set S of a graph is a vertex subset, inducing a κ-regular subgraph such that every vertex not in S has τ neighbors in S. We present a recursive construction of all regular exceptional graphs as successive extensions by regular sets.

Graphs with least eigenvalue −2: Ten years on

“…It turns out that the 3 graphs in the third layer can be constructed as successive (1, 3)-extensions of the line graph 2K 2 . The results are given in the Appendix of [19], where for each graph the list of its immediate successors is given.…”

“…We first describe a recent recursive construction [19] of the regular exceptional graphs, based on the new (κ, τ )-extension technique suggested in [24]. The graphs in question are completely described in [4, Table A3].…”