On the Inverse of Forward Adjacency Matrix

TL;DR

This paper investigates the properties of the inverse of the forward adjacency matrix in lossless LC network graphs, revealing useful physical insights and mathematical properties specific to tree structures.

Contribution

It introduces the forward adjacency matrix, analyzes its inverse, and proves its properties in the context of tree graphs representing lossless LC networks.

Findings

Inverse matrix displays useful physical properties

Matrix is invertible only for tree graphs without cycles

Rigorous proof of matrix properties provided

Abstract

During routine state space circuit analysis of an arbitrarily connected set of nodes representing a lossless LC network, a matrix was formed that was observed to implicitly capture connectivity of the nodes in a graph similar to the conventional incidence matrix, but in a slightly different manner. This matrix has only 0, 1 or -1 as its elements. A sense of direction (of the graph formed by the nodes) is inherently encoded in the matrix because of the presence of -1. It differs from the incidence matrix because of leaving out the datum node from the matrix. Calling this matrix as forward adjacency matrix, it was found that its inverse also displays useful and interesting physical properties when a specific style of node-indexing is adopted for the nodes in the graph. The graph considered is connected but does not have any closed loop/cycle (corresponding to closed loop of inductors in a circuit) as with its presence the matrix is not invertible. Incidentally, by definition the graph being considered is a tree. The properties of the forward adjacency matrix and its inverse, along with rigorous proof, are presented.