The Weighted Complexity and the Determinant Functions of Graphs

Abstract. The complexity of a graph can be obtained as a derivative of a variation of the zeta function [J. Combin. Theory Ser. B, 74 (1998), pp. 408–410] or a partial derivative of its generalized characteristic polynomial evaluated at a point [arXiv:0704.1431[math.CO]]. A similar result for the weighted complexity of weighted graphs was found using a determinant function [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.