Graphs whose normalized Laplacian has three eigenvalues

TL;DR

This paper characterizes graphs with exactly three distinct normalized Laplacian eigenvalues, including known and new exotic examples from combinatorial designs and conference graphs.

Contribution

It provides a combinatorial characterization of graphs with three normalized Laplacian eigenvalues, expanding beyond classical examples.

Findings

Strongly regular and complete bipartite graphs have three normalized Laplacian eigenvalues.

New families of graphs with three eigenvalues are constructed from combinatorial designs.

The paper offers a comprehensive classification of such graphs.

Abstract

We give a combinatorial characterization of graphs whose normalized Laplacian has three distinct eigenvalues. Strongly regular graphs and complete bipartite graphs are examples of such graphs, but we also construct more exotic families of examples from conference graphs, projective planes, and certain quasi-symmetric designs.