Graphs with regular neighbourhoods

Abstract

We call a graph G NR-regular if, for each vertex v of G, the subgraphs of G induced by the vertices adjacent to v and by the vertices not adjacent or equal to v are regular. NR-regular graphs which are regular, disconnected or have disconnected complements are easily classified, and will be called trivial. Those of the first type are just the strongly regular graphs.

We show that non-trivial NR-regular graphs exist and have considerable structure. For example, there are only two vertex degrees, and the vertices of each degree induce a regular subgraph of G. The eigenvalues of each of these subgraphs determine the eigenvalues of the other. We are able to construct non-trivial NR-regular graphs with 4, 8, 28 and 32 vertices, and conjecture that there are infinitely many more.