A complete characterization of graphs with exactly two positive eigenvalues

TL;DR

This paper provides a comprehensive characterization of graphs that have exactly two positive eigenvalues, completing a line of research initiated by Smith's earlier work on graphs with one positive eigenvalue.

Contribution

It offers a complete classification of graphs with exactly two positive eigenvalues, advancing understanding in spectral graph theory.

Findings

Complete characterization of graphs with two positive eigenvalues

Extension of Smith's 1977 results

Clarification of spectral properties of these graphs

Abstract

In 1977 Smith characterized graphs with exactly one positive eigenvalue. Since then, many particular results related to graphs with exactly two positive eigenvalues have emerged. In this paper we conclude this investigation by giving a full characterization of these graphs.