Title: On the Inverse of Forward Adjacency Matrix,
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.