Topology of Clique Complexes of Line Graphs

TL;DR

This paper investigates the topological structure of clique complexes derived from line graphs of various graph classes, providing explicit homotopy types and formulas for these complexes.

Contribution

It determines the homotopy types of clique complexes of line graphs for multiple graph classes and offers closed-form formulas in several cases.

Findings

Homotopy types identified for triangle-free, chordal, and other graph classes.

Explicit formulas for clique complex homotopy types in specific cases.

Enhanced understanding of the topological properties of line graph clique complexes.

Abstract

The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we determine the homotopy type of the clique complexes of line graphs for several classes of graphs including triangle-free graphs, chordal graphs, complete multipartite graphs, wheel-free graphs, and 4-regular circulant graphs. We also give a closed form formula for the homotopy type of these complexes in several cases.