Another construction of edge-regular graphs with regular cliques

TL;DR

This paper presents a novel method for constructing edge-regular graphs with regular cliques that are not strongly regular, including an infinite family and specific examples, and establishes a minimum size for such graphs.

Contribution

It introduces a new construction technique for edge-regular graphs with regular cliques that are not strongly regular, expanding the known classes of such graphs.

Findings

Constructed an infinite family of edge-regular graphs with regular cliques.

Provided an example of an edge-regular graph with parameters (24,8,2).

Proved that graphs with 1-regular cliques that are not strongly regular have at least 24 vertices.

Abstract

We exhibit a new construction of edge-regular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edge-regular graph with parameters $(24,8,2)$. We also show that edge-regular graphs with $1$-regular cliques that are not strongly regular must have at least $24$ vertices.