Second-largest eigenvalue, vertex-connectivity, edge-connectivity, regular multigraph, algebraic connectivity.
The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to connectivity attributes such as the vertex- and edge-connectivity, isoperimetric number, and characteristic path length. In this paper, two upper bounds are presented for the second-largest eigenvalues of regular graphs and multigraphs of a given order which guarantee a desired vertex- or edge-connectivity. The given bounds are in terms of the order and degree of the graphs, and hold with equality for infinite families of graphs. These results answer a question of Mohar.
Abiad, Aida; Brimkov, Boris; Martinez-Rivera, Xavier; O, Suil; and Zhang, Jingmei.
"Spectral Bounds for the Connectivity of Regular Graphs with Given Order",
Electronic Journal of Linear Algebra,
Volume 34, pp. 428-443.