Cataloguing general graphs by point and line spectra Original Research Article

Certain perturbative expansions in physics are sums over all multigraphs with loops — i.e., graph-like structures in which each point can be connected to any other point, including itself, by an arbitrary number of lines. We treat the problem of exhibiting, or at least counting, all such objects for all reasonably small numbers of points and lines, as well as the associated problem of determining the number of labeled graphs corresponding to a given unlabeled one. Symbolic computation (specifically, Mathematica) has proven very useful here. It is helpful to classify the graphs further according to the distribution of the lines among the pairs of points (line spectrum) and simultaneously the distribution of the ends of the lines among the points (point spectrum). The basic tool is a generating function in which each coefficient is the number of general graphs with a given line spectrum and point spectrum.