Number of Subgraphs and Their Converses in Tournaments and New Digraph Polynomials

TL;DR

This paper introduces a digraph polynomial to characterize converse invariance in oriented graphs, providing new results on trees, regular graphs, and general connected graphs, and proposing a related conjecture.

Contribution

It develops a new polynomial tool to identify converse invariance and characterizes certain classes of graphs with this property, advancing understanding in graph theory.

Findings

Characterized all orientations of trees with diameter at most 3 that are converse invariant.

Proved that all orientations of regular graphs are not converse invariant if degree sequences differ.

Showed that connected graphs with maximum degree at least 3 can have orientations that are not converse invariant.

Abstract

An oriented graph $D$ is converse invariant if, for any tournament $T$, the number of copies of $D$ in $T$ is equal to that of its converse $-D$. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph $D$ with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all converse invariant oriented graphs? In this paper, we introduce a digraph polynomial and employ it to give a necessary condition for an oriented graph to be converse invariant. This polynomial serves as a cornerstone in proving all the results presented in this paper. In particular, we characterize all orientations of trees with diameter at most 3 that are converse invariant. We also show that all orientations of regular graphs are not converse invariant if $D$ and $-D$ have different degree sequences. In addition, in contrast to the findings of El Sahili and Ghazo Hanna, we prove that every connected graph $G$ with maximum degree at least $3$, admits an orientation $D$ of $G$ such that $D$ is not converse invariant. We pose one conjecture.