We introduce the notion of star cluster of a simplex in a simplicial complex. This concept provides a general tool to study the topology of independence complexes of graphs. We use star clusters to answer a question arisen from works of Engström and Jonsson on the homotopy type of independence complexes of triangle-free graphs and to investigate a large number of examples which appear in the literature. We present an alternative way to study the chromatic and clique numbers of a graph from a homotopical point of view and obtain new results regarding the connectivity of independence complexes. © 2013 Elsevier Ltd.
Barmak, J. A. (2013). Star clusters in independence complexes of graphs. Advances in Mathematics, 241, 33–57. https://doi.org/10.1016/j.aim.2013.03.016