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arXiv · 1703.02748

Spectral Bounds for the Connectivity of Regular Graphs with Given Order

Aida Abiad·Boris Brimkov·Xavier Martinez-Rivera·O Suil·Jingmei Zhang
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Abstract

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, we present two upper bounds 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.

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Aida Abiad, Boris Brimkov, Xavier Martinez-Rivera, O Suil, Jingmei Zhang. 2017-03-08. Spectral Bounds for the Connectivity of Regular Graphs with Given Order. https://arxiv.org/abs/1703.02748

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