The weighted complexity and the determinant functions of graphs

TL;DR

This paper explores the relationship between graph complexity, determinant functions, and zeta functions, providing new formulas for weighted complexities and applying them to graph products.

Contribution

It introduces a new determinant function approach to express weighted graph complexities and derives formulas for a novel weighted complexity involving spanning tree weights.

Findings

Derived a condition linking weighted complexity to the determinant function.

Obtained a new formula for the Bartholdi zeta function.

Computed weighted complexities for products of complete graphs.

Abstract

The complexity of a graph can be obtained as a derivative of a variation of the zeta function or a partial derivative of its generalized characteristic polynomial evaluated at a point [\textit{J. Combin. Theory Ser. B}, 74 (1998), pp. 408--410]. A similar result for the weighted complexity of weighted graphs was found using a determinant function [\textit{J. Combin. Theory Ser. B}, 89 (2003), pp. 17--26]. In this paper, we consider the determinant function of two variables and discover a condition that the weighted complexity of a weighted graph is a partial derivative of the determinant function evaluated at a point. Consequently, we simply obtain the previous results and disclose a new formula for the Bartholdi zeta function. We also consider a new weighted complexity, for which the weights of spanning trees are taken as the sum of weights of edges in the tree, and find a similar formula for this new weighted complexity. As an application, we compute the weighted complexities of the product of the complete graphs.