Eigenvalues of the topological matrix. Splitting of graphs with symmetrical components and alternant graphs

Abstract

If a topological matrix M is represented by a graph G then unitary transformation of M to the block-diagonal diag {M₁, M₂, …} is equivalent to splitting G into disconnected sub-graphs G₁, G₂, … such that graph G₁ represents M₁, etc. The eigenvalues of M are those of M₁, M₂, … separately, and such graph-splitting permits the matrices M₁, M₂… to be set up by inspection. In the present paper algorithms are given for effecting this graph-splitting for (1) systems which may or may not be symmetrical as a whole but which contain fragments having two-fold symmetry, and (2) alternant systems.