Spectral bounds for the connectivity of regular graphs with given order

Aida Abiad; Boris Brimkov; Xavier Martínez-Rivera; O. Suil; Jingmei Zhang

doi:10.13001/1081-3810.3675

Spectral bounds for the connectivity of regular graphs with given order

Aida Abiad, Boris Brimkov, Xavier Martínez-Rivera, O. Suil, Jingmei Zhang

Research output: Contribution to journal › Article › Academic › peer-review

10 Citations (Scopus)

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, 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.

Original language	English
Article number	33
Pages (from-to)	428-443
Number of pages	16
Journal	Electronic Journal of Linear Algebra
Volume	34
DOIs	https://doi.org/10.13001/1081-3810.3675
Publication status	Published - Sept 2018
Externally published	Yes

Keywords

Algebraic connectivity
Edge-connectivity
Regular multigraph
Second-largest eigenvalue
Vertex-connectivity

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10.13001/1081-3810.3675

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author = "Aida Abiad and Boris Brimkov and Xavier Mart{\'i}nez-Rivera and O. Suil and Jingmei Zhang",

year = "2018",

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T1 - Spectral bounds for the connectivity of regular graphs with given order

AU - Abiad, Aida

AU - Brimkov, Boris

AU - Martínez-Rivera, Xavier

AU - Suil, O.

AU - Zhang, Jingmei

PY - 2018/9

Y1 - 2018/9

N2 - 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.

AB - 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.

KW - Algebraic connectivity

KW - Edge-connectivity

KW - Regular multigraph

KW - Second-largest eigenvalue

KW - Vertex-connectivity

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JO - Electronic Journal of Linear Algebra

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Spectral bounds for the connectivity of regular graphs with given order

Abstract

Keywords

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