A study on $T$-equivalent graphs

TL;DR

This paper extends Tutte's results on $T$-equivalent graphs by allowing larger rotors and introduces a new method to generate infinitely many non-isomorphic $T$-equivalent graph pairs.

Contribution

The paper generalizes Tutte's theorem to rotors of order at least 6 under certain conditions and presents a novel approach for creating infinitely many non-isomorphic $T$-equivalent graphs.

Findings

Extension of Tutte's $T$-equivalence result to larger rotors

Introduction of a new method for generating $T$-equivalent graph pairs

Identification of conditions for $T$-equivalence with rotors of order ≥ 6

Abstract

In his article [J. Comb. Theory Ser. B 16 (1974), 168-174], Tutte called two graphs $T$-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs $G$ and $G'$ are $T$-equivalent if $G'$ is obtained from $G$ by flipping a rotor (i.e., replacing it by its mirror) of order at most $5$, where a rotor of order $k$ in $G$ is an induced subgraph $R$ having an automorphism $\psi$ with a vertex orbit $\{\psi^i(u): i\ge 0\}$ of size $k$ such that every vertex of $R$ is only adjacent to vertices in $R$ unless it is in this vertex orbit. In this article, we first show the above result due to Tutte can be extended to a rotor $R$ of order $k\ge 6$ if the subgraph of $G$ induced by all those edges of $G$ which are not in $R$ satisfies certain conditions. Also, we provide a new method for generating infinitely many non-isomorphic $T$-equivalent pairs of graphs.