The algebraic connectivity of barbell graphs

The algebraic connectivity of a graph is the second smallest eigenvalue of its Laplacian matrix. An eigenvector affording the algebraic connectivity is called a Fiedler vector. The barbell graph Bp,q;l is the graph obtained by joining a vertex in a cycle Cp(p≠2) and a vertex in a cycle Cq(q≠2) by a path Pl with p≥3 or q≥3, and l≥2 if p=1 or q=1. In this paper, we determine the graphs minimizing the algebraic connectivity among all barbell graphs and the graphs containing a barbell graph as a spanning subgraph of given order, respectively. Moreover, we investigate how the algebraic connectivity behaves under some graph perturbations, and compare the algebraic connectivities of barbell graphs, cycles, and θ-graphs.