Acyclic polynomials of graphs

TL;DR

This paper introduces the acyclic polynomial of a graph, explores its properties, and studies the roots' nature and location, providing insights into the combinatorial structure of acyclic subgraphs.

Contribution

It defines the acyclic polynomial for graphs, analyzes its properties, and investigates the roots' behavior, offering new understanding of graph acyclicity enumeration.

Findings

Properties of acyclic polynomials are characterized.

Roots of the acyclic polynomial are analyzed for various graph classes.

Insights into the distribution and location of roots are provided.

Abstract

For each nonnegative integer $i$, let $a_i$ be the number of $i$-subsets of $V(G)$ that induce an acyclic subgraph of a given graph $G$. We define $A(G,x) = \sum_{i \geq 0} a_i x^i$ (the generating function for $a_i$) to be the acyclic polynomial for $G$. After presenting some properties of these polynomials, we investigate the nature and location of their roots.