Computing the Characteristic polynomial of a graph

Abstract

How can one actually compute the eigenvalues of a graph? In principal, there are three methods. Namely, (1) we can search for p orthogonal eigenvectors, (2) we can determine the first p moments by counting closed walks and then find the spectrum from the moments, or (3) we can use certain subgraphs to determine the coefficients of the characteristic polynomial and then find its roots.

In practice, however, all of these approaches may prove to be too tedious. If the spectrum is a natural concept, then we should expect to find simple relations between the spectra of related graphs. This is indeed the case. In Sections 2 and 3, we present four theorems which are structural results linking the spectrum of a graph to the spectra of certain subgraphs. Then in Section 4, we find the spectra of graphs formed by certain binary operations. In the final section, we apply these results to obtain the spectra of several known families of graphs: complete graphs, complete bigraphs, cubes, cycles, wheels, paths, ladders, and möbius ladders.

Research supported in part by grant 73-2502 from the Air Force Office of Scientific Research. This paper was part of the author's doctoral dissertation "The Spectrum of a Graph," The University of Michigan, 1973.