Graph Polynomial for Colored Embedded Graphs: A Topological Approach

TL;DR

This paper introduces a topologically inspired polynomial invariant for embedded graphs, extending it to colored graphs, analyzing its behavior under graph operations, and exploring applications in graph detection and topological entropy.

Contribution

It develops a novel algebraic topological polynomial for embedded graphs, including colored variants, and investigates its properties and applications in graph theory and physics.

Findings

Polynomial detects certain classes of graphs

Connects to topological entanglement entropy

Describes polynomial changes under graph operations

Abstract

We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in physics. We also analyze a variant of these polynomials for colored embedded graphs. This is used to describe the change in the polynomial under basic graph theoretic operations. We conclude with several applications of this polynomial including detection of certain classes of graphs and the connection of this polynomial with topological entanglement entropy.