# Topological and Algebraic Characterizations of Gallai-Simplicial Complexes

## TL;DR

This paper explores the topological and algebraic properties of Gallai-simplicial complexes derived from planar graphs, computing Euler characteristics for specific graph classes and characterizing when certain graphs are Gallai graphs.

## Contribution

It provides new formulas for Euler characteristics of Gallai-simplicial complexes and characterizes classes of graphs that are Gallai graphs based on their construction.

## Key findings

- Euler characteristics computed for triangular ladder and prism graphs.
- Identification of conditions under which specific graphs are Gallai graphs.
- New algebraic and topological characterizations of Gallai-simplicial complexes.

## Abstract

We recall first Gallai-simplicial complex $\Delta_{\Gamma}(G)$ associated to Gallai graph $\Gamma(G)$ of a planar graph $G$. The Euler characteristic is a very useful topological and homotopic invariant to classify surfaces. In Theorems 3.2 and 3.4, we compute Euler characteristics of Gallai-simplicial complexes associated to triangular ladder and prism graphs, respectively. Let $G$ be a finite simple graph on $n$ vertices of the form $n=3l+2$ or $3l+3$. In Theorem 4.4, we prove that $G$ will be $f$-Gallai graph for the following types of constructions of $G$. Type 1. When $n=3l+2$. $G=\mathbb{S}_{4l}$ is a graph consisting of two copies of star graphs $S_{2l}$ and $S'_{2l}$ with $l\geq 2$ having $l$ common vertices. Type 2. When $n=3l+3$. $G=\mathbb{S}_{4l+1}$ is a graph consisting of two star graphs $S_{2l}$ and $S_{2l+1}$ with $l\geq 2$ having $l$ common vertices.