On the limit points of the smallest positive eigenvalues of graphs

Abstract

In 1972, Hoffman [11] initiated the study of limit points of eigenvalues of nonnegative symmetric integer matrices. He posed the question of finding all limit points of the set of spectral radii of all nonnegative symmetric integer matrices. In the same article, the author demonstrated that it is enough to consider the adjacency matrices of simple graphs to study the limit points of spectral radii. Since then, many researchers have worked on similar problems, considering various specific eigenvalues such as the least eigenvalue, the kth largest eigenvalue, and the kth smallest eigenvalue, among others. Motivated by this, we ask the question, “which real numbers are the limit points of the set of the smallest positive eigenvalues (respectively, the largest negative eigenvalues) of graphs?” In this article, we provide a complete answer to this question by proving that any nonnegative (respectively, nonpositive) real number is a limit point of the set of all smallest positive eigenvalues (respectively, largest negative eigenvalues) of graphs. We also show that the union of the sets of limit points of the smallest positive eigenvalues and the largest negative eigenvalues of graphs is dense in $R$, the set of all real numbers.