The weighted complexity and the determinant functions of graphs

Details The weighted complexity and the determinant functions of graphs The weighted complexity and the determinant functions of graphs

2007-05-16

arxiv.org/abs/0705.2284v2

Linear algebra and its applications, 433 (2010), 348-355

Mathematics

Combinatorics

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

The weighted complexity and the determinant functions of graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The weighted complexity and the determinant functions of graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The weighted complexity and the determinant functions of graphs will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-670055