On Clique Graphs and Clique Regular Graphs

TL;DR

This paper explores clique graphs derived from graphs where each edge belongs to a unique maximum clique, providing bounds on eigenvalues, classifications for special cases, and applications to problems like Conway's 99-graph problem.

Contribution

It introduces new theoretical results on clique graphs, including eigenvalue bounds and classifications for strongly regular graphs, expanding understanding of these structures.

Findings

Eigenvalue bounds for clique graphs, especially for k-regular graphs

Complete classifications for strongly regular graphs

Applications to Conway's 99-graph problem

Abstract

If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general results and classifications related to these clique graph that will be useful to researchers studying these objects. In particular, we find bounds on its eigenvalues (with exact results when $\Gamma$ is $k$-regular) and some complete classifications when $\Gamma$ is strongly regular. We apply our results to many examples, including Conway's 99-graph problem and the existence problem for other strongly regular graphs.