Unbalanced signed bicyclic graphs minimizing the least eigenvalue

Abstract

Given a signed graph G˙, let A(G˙) denote its adjacency matrix. The eigenvalues of A(G˙) are called the eigenvalues of G˙. In this study we focus on the least eigenvalues of connected signed graphs, and consider the behavior of the least eigenvalue of a signed graph when it is perturbed by some edge variations. In particular, we identify the first four smallest eigenvalues with their extremal signed graphs among all the unbalanced signed bicyclic graphs of order n≥31 and range them corresponding to their least eigenvalues in an increasing order.