On the extreme complexity of certain nearly regular graphs

TL;DR

This paper investigates the complexity of certain nearly regular graphs, showing that specific strongly regular graphs have maximal complexity, with implications for biological neuronal activity modeling.

Contribution

It demonstrates that known triangle-free strongly regular graphs have maximal complexity and generalizes these results to nearly regular graphs with two Laplacian eigenvalues.

Findings

Seven known triangle-free strongly regular graphs are of maximal complexity.

Their complements are of minimal complexity.

Generalization to nearly regular graphs with two Laplacian eigenvalues.

Abstract

The complexity of a graph is the number of its labeled spanning trees. It is demonstrated that the seven known triangle-free strongly regular graphs, such as the Higman-Sims graph, are graphs of maximal complexity among all graphs of the same order and degree; their complements are shown to be of minimal complexity. A generalization to nearly regular graphs with two distinct eigevalues of the Laplacian is presented. Conjectures and applications of these results to biological problems on neuronal activity are described.