The Tutte symmetric matrix of a graph

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).

Figures (8)

• Figure 1: Graphs $G$ and $H$ glued at a single vertex.
• Figure 2: The subgraph triples used to calculate $XB_{P_2}(\bm x; t)$.
• Figure 3: The subgraph triples used to calculate $M_{P_2}$.
• Figure 4: Example of map $\varphi$ with $i=1, j=2, k=2$.
• Figure 5: The graph $S_4=K_{1,3}$ obtained by gluing three $P_2$ graphs at a common vertex $x$.

Theorems & Definitions (60)

• Example 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Example 2.4
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Proposition 3.4