Minimizing the Least Eigenvalue of Bicyclic Graphs with Fixed Diameter

Abstract

Let \(\mathcal{B}(n,d)\) be the set of bicyclic graphs with both \(n\) vertices and diameter \(d\), and let \(\theta^*\) consist of three paths \(u_0w_1v_0\), \(u_0w_2v_0\), and \(u_0w_3v_0\). For four nonnegative integers \(n,d,k,j\) satisfying \(n \geq d+3\), \(d=k+j+2\), we let \(B(n,d;k,j)\) denote the bicyclic graph obtained from \(\theta^*\) by attaching a path of length \(k\) to \(u_0\), attaching a path of length \(j\) to vertex \(v_0\) and \(n-d-3\) pendant edges to \(w_0\), and let \(\mathcal{B}(n,d;k,j) = \{B(n,d;k,j) \mid k+j \geq 1\}\). In this paper, the extremal graphs with the minimal least eigenvalue among all graphs in \(\mathcal{B}(n,d;k,j)\) are well characterized, and some structural characterizations about the extremal graphs with the minimal least eigenvalue among all graphs in \(\mathcal{B}(n,d)\) are presented as well.