Group Inverse of the Laplacian of Connections of Networks

TL;DR

This paper derives formulas relating the group inverse of a network formed by connecting two networks with new edges to the group inverses of the original networks, including a specific case with one edge and the Kirchhoff index.

Contribution

It introduces new formulas for the group inverse of combined networks based on the original networks' group inverses, expanding previous perturbation-based results.

Findings

Derived formula for connecting two networks with a single edge.

Established relation for the Kirchhoff index of the combined network.

Extended previous work on network perturbations to network connections.

Abstract

In previous works the group inverse of a network obtained by some perturbations, as the deletion of a vertex, the addition of a new vertex, contraction of an edge, etc. is obtained in terms of the group inverse of the original network. In this work, two given networks are connected with some new edges and the group inverse of the new network is related with the group inverses of the two original networks. In particular, the formula for the connection of two networks by just one edge is obtained, and besides the formula for the Kirchhoff index of this kind of network.