The Least Eigenvalues of Unicyclic Graphs

Abstract

Let \(G\) be a unicyclic graph on \(n \geq 3\) vertices. Let \(A(G)\) be the adjacency matrix of \(G\). The eigenvalues of \(A(G)\) are denoted by \(\lambda_1(G) \geq \lambda_2(G) \geq \cdots \geq \lambda_n(G)\), which are called the eigenvalues of \(G\). Let the unicyclic graphs \(G\) on \(n\) vertices be ordered by their least eigenvalues \(\lambda_n(G)\) in non-decreasing order. For \(n \geq 14\), the first six graphs in this order are determined.