Signed a-polynomials of graphs and Poincar\'e polynomials of real toric manifolds

TL;DR

This paper introduces a signed a-polynomial invariant for graphs, generalizing signed a-numbers, and relates it to Poincaré polynomials of real toric manifolds, providing formulas for various graph classes and their topological invariants.

Contribution

It defines the signed a-polynomial, connects it to Poincaré polynomials of real toric manifolds, and derives generating functions and topological invariants for multiple graph types.

Findings

Generated explicit formulas for signed a-polynomials of various graphs.

Calculated Euler characteristics and Betti numbers for real toric manifolds of complete multipartite graphs.

Established relationships between graph invariants and topological properties of associated manifolds.

Abstract

Recently, Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph $G$ is related to the Poincar\'e polynomial $P_{M(G)}(z)$, which is the generating function for the Betti numbers of the real toric manifold $M(G)$. We give the generating functions for the signed a-polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold $M(G)$ for complete multipartite graphs $G$.