Graph-counting polynomials for oriented graphs

TL;DR

This paper investigates the properties of graph-counting polynomials for oriented graphs, focusing on the zeros' locations when counting specific subgraphs, such as unbranched subgraphs.

Contribution

It extends the study of graph-counting polynomials to oriented graphs and identifies conditions affecting the zeros' locations for certain subgraph classes.

Findings

Identifies cases where zeros of the polynomial are confined to specific regions

Analyzes the impact of subgraph types on polynomial zeros

Provides theoretical insights into polynomial zero distributions

Abstract

If ${\cal F}$ is a set of subgraphs $F$ of a finite graph $E$ we define a graph-counting polynomial $$ p_{\cal F}(z)=\sum_{F\in{\cal F}}z^{|F|} $$ In the present note we consider oriented graphs and discuss some cases where ${\cal F}$ consists of unbranched subgraphs $E$. We find several situations where something can be said about the location of the zeros of $p_{\cal F}$.