Vertex-transitive Neumaier graphs

Abstract

A graph $Γ$ is called edge-regular whenever it is regular and for any two adjacent vertices, the number of their common neighbors is independent of the choice of vertices. A clique $C$ in $Γ$ is called regular whenever for any vertex out of $C$, the number of its neighbors in $C$ is independent of the vertex. A Neumaier graph is a non-complete edge-regular graph with a regular clique. In this paper, we study vertex-transitive Neumaier graphs. We give a necessary and sufficient condition under which a vertex-transitive Neumaier graph is strongly regular. We also identify Neumaier Cayley graphs with small valency at most $10$ among vertex-transitive Neumaier graphs.

Figures (1)

• Figure 1: A Neumaier graph with parameters $(n,k,\lambda;a,c)$. Let $C \subset V$ denote a regular clique with nexus $a$ containing an arbitrary element $e$ and $S$ denote the set of neighbors of $e$. The notation on the arrow between two partitions is the number of adjacent vertices from one vertex in a partition to the vertices of another partition. The notation in a partition is the valency of the regular induced subgraph on that partition.

Theorems & Definitions (50)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof