Abstract
This paper aims to study a family of distances in networks associated with effective resistances. Specifically, we consider the effective resistance distance with respect to a positive parameter and a weight on the vertex set; that is, the effective resistance distance associated with an irreducible and symmetric M-matrix whose lowest eigenvalue is the parameter and the weight function is the associated eigenfunction. The main idea is to consider the network embedded in a host network with additional edges whose conductances are given in terms of the mentioned parameter. The novelty of these distances is that they take into account not only the influence of shortest and longest weighted paths but also the importance of the vertices. Finally, we prove that the adjusted forest metric introduced by P. Chebotarev and E. Shamis is nothing else but a distance associated with a Schr?dinger operator with constant weight.
Highlights
Because of its structural meaning, the resistance distance has become a useful tool to analyze structural properties of graphs, or more generally of networks, such as robustness, see for instance [13]
The high sensibility of this metric with respect to small perturbations, makes it suitable to compare different network structures. This is one of the main reasons for which effective resistances and the corresponding Kirchhoff Index, the sum of all of them, have emerged as a structure–descriptor in the framework of Organic Chemistry, where the topology of chemical compounds is conventionally represented by a molecular network where edge weights correspond to bond properties, see for instance [15] and references therein
For a fixed weight function, we prove that the associated effective resistance distances are continuous and monotone decreasing with respect to the parameter and they are upper bounded by the weighted geodesic distance of the network
Summary
Because of its structural meaning, the resistance distance has become a useful tool to analyze structural properties of graphs, or more generally of networks, such as robustness, see for instance [13]. We prove that for each weight function, the corresponding one parametric family of resistance distances can be understood as the effective resistance on a host network associated with a singular positive semi–definite Schrodinger operator. For any network with constant weight on the vertices, the one parametric family of distances corresponds to the effective resistance associated with symmetric and diagonally dominant M –matrices with constant diagonal excess. We show that these resistance distances coincide with the so–called adjusted forest metrics introduced by P. They interpret these metrics as a measure of the accessibility from a vertex to another and they form a one–parametric family of distances, where the parameter determines the proportion of taking into account long and short routes between vertices
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