On Clique Graphs and Clique Regular Graphs

TL;DR

This work introduces clique graphs $C_(Γ)$ as a broad generalization of line graphs, along with the notion of $$-clique regular graphs. It establishes a precise spectral relationship between a regular graph $Γ$ and its $$-clique graph, and provides conditions under which $C_(Γ)$ is itself strongly regular. The paper then applies these results to diverse graph families—OA block graphs, triangular graphs, generalized quadrangles, and locally linear graphs—deriving explicit structures and spectra, with applications to longstanding existence problems such as Conway's 99-graph problem. Overall, it offers new tools for spectral analysis and classification of clique-based graph constructions across several canonical graph families.

Abstract

If $Γ$ is a graph for which every edge is in exactly one clique of order $ω$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $Γ$. 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 $Γ$ is $k$-regular) and some complete classifications when $Γ$ 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.

Figures (11)

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Theorems & Definitions (59)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Remark 3.5
• Theorem 3.6