On the clique behavior of graphs of low degree

TL;DR

This paper investigates the behavior of iterated clique graphs of low-degree graphs, proving that the octahedron is the only divergent connected graph with maximum degree 4.

Contribution

It establishes that among connected graphs with maximum degree 4, only the octahedron exhibits divergent clique graph behavior.

Findings

The octahedron is the only divergent connected graph with degree 4.

Connected graphs with degree less than 4 are all convergent.

The result narrows the class of divergent graphs in low-degree graphs.

Abstract

To any simple graph $G$, the clique graph operator $K$ associates the graph $K(G)$ which is the intersection graph of the maximal complete subgraphs of $G$. The iterated clique graphs are defined by $K^{0}(G)=G$ and $K^{n}(G)=K(K^{n-1}(G))$ for $n\geq 1$. If there are $m<n$ such that $K^{m}(G)$ is isomorphic to $K^{n}(G)$ we say that $G$ is convergent, otherwise, $G$ is divergent. The first example of a divergent graph was shown by Neumann-Lara in the 1970s, and is the graph of the octahedron. In this paper, we prove that among the connected graphs with maximum degree 4, the octahedron is the only one that is divergent.