Alexander polynomial and spanning trees

TL;DR

This paper demonstrates that the Alexander polynomial evaluated at t=1 equals the weighted count of spanning trees in a class of spatial graphs, extending classical knot invariants to graph theory.

Contribution

It establishes a new connection between the Alexander polynomial and spanning trees in weighted spatial graphs, generalizing previous combinatorial constructions.

Findings

Alexander polynomial at t=1 equals weighted spanning tree count

Extends classical knot invariants to spatial graph theory

Provides a combinatorial interpretation of the polynomial value

Abstract

Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander polynomial evaluated at $t=1$ gives the weighted number of the spanning trees of the graph.