On a Problem of Harary and Schwenk on Graphs with Distinct Eigenvalues

TL;DR

This paper provides algebraic characterizations for graphs with distinct eigenvalues, addressing a longstanding problem by Harary and Schwenk through necessary and sufficient conditions involving Hermitian and positive semidefinite matrices.

Contribution

It introduces new algebraic criteria for graphs with distinct eigenvalues, extending the problem to Hermitian and Laplacian matrices with simple spectral properties.

Findings

Characterization of graphs with distinct adjacency eigenvalues

Conditions for Hermitian matrices with simple spectral radius

Criteria for Laplacian matrices with simple least eigenvalue

Abstract

Harary and Schwenk posed the problem forty years ago: Which graphs have distinct adjacency eigenvalues? In this paper, we obtain a necessary and sufficient condition for an Hermitian matrix with simple spectral radius and distinct eigenvalues. As its application, we give an algebraic characterization to the Harary-Schwenk's problem. As an extension of their problem, we also obtain a necessary and sufficient condition for a positive semidefinite matrix with simple least eigenvalue and distinct eigenvalues, which can provide an algebraic characterization to their problem with respect to the (normalized) Laplacian matrix.