The orderings of bicyclic graphs and connected graphs by algebraic connectivity
Summary: The algebraic connectivity of a graph \(G\) is the second smallest eigenvalue of its Laplacian matrix. Let \({\mathcal B}_n\) be the set of all bicyclic graphs of order \(n\). In this paper we determine the last four bicyclic graphs (according to their smallest algebraic connectivities) among all graphs in \({\mathcal B}_n\) when \(n\geq 13\). This result, together with our previous results on trees and unicyclic graphs, can be used to further determine the last sixteen graphs among all connected graphs of order \(n\). This extends the results of \textit{J.-Y. Shao, J.-M. Guo} and \textit{H.-Y. Shan} [Linear Algebra Appl. 428, No. 7, 1421--1438 (2008; Zbl 1134.05063)].
scientific article; zbMATH DE number 5827411