# Characterizations of Line Simplicial Complexes

## TL;DR

This paper explores the properties of line simplicial complexes derived from graphs, establishing their connectedness, relationships with Euler characteristics, and shellability for different graph classes.

## Contribution

It introduces and characterizes line and Gallai simplicial complexes, proving their connectedness criteria and analyzing their topological properties such as shellability.

## Key findings

- Line simplicial complex is connected iff the graph is connected.
- Euler characteristics relate line and Gallai simplicial complexes.
- Shellability varies across different classes of graphs.

## Abstract

Let $G$ be a finite simple graph. The line graph $L(G)$ represents the adjacencies between edges of $G$. We define first the line simplicial complex $\Delta_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes $\Delta_{\Gamma}(G)$ and $\Delta_{\Gamma'}(G)$ (respectively) as spanning subcomplexes. The study of connectedness of simplicial complexes is interesting due to various combinatorial and topological aspects. In Theorem 3.3, we prove that the line simplicial complex $\Delta_L(G)$ is connected if and only if $G$ is connected. In Theorem 3.4, we establish the relation between Euler characteristics of line and Gallai simplicial complexes. In Section 4, we discuss the shellability of line and anti-Gallai simplicial complexes associated to various classes of graphs.