Golden Laplacian Graphs

Sadia Akhter, Mattia Frasca and Ernesto Estrada ()
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Sadia Akhter: Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
Mattia Frasca: DIEEI, University of Catania, via Santa Sofia 64, 95125 Catania, Italy
Ernesto Estrada: Institute for Cross-Disciplinary Physics and Complex Systems (IFISC), CSIC-UIB, 07122 Palma de Mallorca, Spain

Mathematics, 2024, vol. 12, issue 4, 1-21

Abstract: Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes n ≤ 10 . We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks.

Keywords: Laplacian matrix; eigenvalues; golden ratio; Laplacian eigenratio; synchronizability; algebraic graph theory; robustness (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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