The dependence of graph energy on network structure

4th International Workshop on Statistical Physics and Mathematics for Complex Systems

物理学;数学

Zhu, Y.^1,2,3 ; Li, W.^2,4 ; Wang, Q.A.^1,3 ; Cai, X.^2

ISMANS, LUNAM Université, 44 av. Bartholdi, Le Mans

72000, France^1

Complexity Science Center, Institute of Particle Physics, Central China Normal University, Wuhan

430079, China^2

IMMM, UMR CNRS 6283, Université du Maine, Le Mans

72085, France^3

Max-Planck Institute for Mathematics in the Sciences, Inselst. 22, Leipzig

04103, Germany^4

关键词: Adjacency matrices;    Degree distributions;    Degree sequence;    Energy of a graph;    Generating methods;    Network structures;    Power-law functions;    Topological structure;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/604/1/012024/pdf
DOI  :  10.1088/1742-6596/604/1/012024

来源: IOP

【 摘 要 】

In this paper, we aim at investigating how the energy of a graph depends upon its underlying topological structure for regular and sparse scale free networks. Firstly, the spectra and energies of some simple regular graphs are calculated exactly and an exact expression is derived for the eigenvalues of adjacency matrix of regular graphs with degree k being given by k = 2a(a = 1, 2, 3,). It is also found that a graph with k being about 0.8N owns the largest energy for the regular graphs with the same size and the same generating method used in this paper. Furthermore, we investigate the energy of sparse scale-free networks with different average degreeand degree distribution exponent γ. While γ is specified, the energy is a power-law function ofwith exponent being about 0.5. And whileis fixed, energy will be obviously proportional to γ. Otherwise, we also find that the energy is a power- law function of the variance of degree sequence with exponent weakly depending on the size of network. Interestingly, while bothand γ are specified, there will be a terrific linear fit to the relationship between energy and the size of system.