The Tutte symmetric matrix of a graph

TL;DR

This paper introduces a matrix-based formula for the Tutte symmetric function of a graph, linking graph operations to matrix operations such as multiplication, trace, and transpose.

Contribution

It provides a novel matrix representation for the Tutte symmetric function, enabling easier computation and understanding of graph operations.

Findings

Matrix $M_G$ recovers the Tutte symmetric function.

Graph gluing corresponds to matrix multiplication.

Reversing a graph corresponds to matrix transpose.

Abstract

We provide a matrix-based formula for the Tutte symmetric function of a graph. In particular, for any graph $G$ with a designated head and tail vertex, we describe an infinite matrix $M_G$ from which the Tutte symmetric function can be easily recovered. We prove gluing graphs together corresponds to matrix multiplication, gluing the head and tail of a single graph corresponds to taking the trace, and reversing a graph corresponds to the transpose (up to a change of basis).