Line graphs of simplicial complexes

TL;DR

This paper explores the properties of line graphs derived from pure simplicial complexes, establishing connections between their combinatorial structure and algebraic invariants like Betti numbers, and characterizing specific classes of these complexes.

Contribution

It introduces a method to compute the second graded Betti number of facet ideals using the line graph's structure and characterizes complexes with complete or bipartite line graphs.

Findings

Betti number computation from line graph structure

Characterization of complexes with complete line graphs

Conditions for line graphs of simplicial complexes

Abstract

We consider the line graph of a pure simplicial complex. We prove that, as in the case of line graphs of simple graphs, one can compute the second graded Betti number of the facet ideal of a pure simplicial complex in terms of the combinatorial structure of its line graph. We characterize those pure simplicial complexes whose line graph is a complete (bipartite) graph. We give conditions that line graphs of simplicial complexes should fulfill.