Research and Application of Hypernetwork Energy

doi:10.3778/j.issn.1673-9418.2006065

Abstract

Abstract:

Graph energy plays an important role in research of graph theory. Graph energy and many other similar variants have been applied in many other types of graphs, e.g., undirected graphs, oriented graphs, mixed graphs, and so on. Hypernetwork is a kind of network which is more complex than traditional complex network. Most graph energies are calculated based on the eigenvalues of matrix and can hardly be extended to hypernetwork, which limits their application range. Network energy based on network dimension has been applied in undirected graphs and oriented graphs one by one, and is closely related to many other energy-like quantities, e.g., graph energy of undirected graphs, skew energy of oriented graphs, and so on. In this paper, based on hypernetwork dimension of hypernetwork and combined with network energy, hypernetwork energy of hypernetwork is proposed with application of hypernetwork dimension on hypernetwork. At the same time, several upper and lower limits of hypernetwork energy are given, and the internal relations between hypernetwork energy of hypernetwork and network energy of graphs are demonstrated. Finally, several important properties of hypernetwork energy are analyzed as well.

Key words: complex network, hypernetwork, graph energy, network energy, hypernetwork energy

摘要：

图能量是图论研究的重要内容，图能量及其变种已在无向图、有向图、混合图等其他多种类型的图中得到很多成功的应用。超网络是一类较传统意义上的复杂网络更为复杂的网络。大多数图能量均是基于矩阵特征值计算得到的，无法推广应用到超网络中，应用范围受限。基于网络维数的网络能量已先后应用于无向图、有向图等多种类型图的分析研究中，并与无向图的图能量及有向图的斜能量等其他类似能量之间存在密切关联。基于超网络的超网络维数，结合网络能量，将超网络维数应用于超网络中，提出了超网络的超网络能量，给出了超网络能量的若干上下限，论证了超网络的超网络能量与图的网络能量之间的内在关联，最后分析了超网络的超网络能量若干重要性质。

关键词: 复杂网络, 超网络, 图能量, 网络能量, 超网络能量

LIU Shengjiu, LI Tianrui, LIU Jia, XIE Peng. Research and Application of Hypernetwork Energy[J]. Journal of Frontiers of Computer Science and Technology, 2021, 15(4): 682-689.

刘胜久, 李天瑞, 刘佳, 谢鹏. 超网络能量研究与应用[J]. 计算机科学与探索, 2021, 15(4): 682-689.

References

[1] ERDOS P, HAJNAL A. On the chromatic number of graphs and set systems[J]. Acta Mathematica Academiae Scientiarum Hungarica, 1966, 17: 61-99.
[2] BERGE C. Graphs and hypergraphs[M]. Amsterdam: North-Holland Publishing Company, 1973.
[3] ESTRADA E, RODRIGUES V R. Subgraph centrality in complex networks[J]. Physical Review E, 2005, 71(5): 1-9.
[4] LIU S J, LI T R, HORNG S J, et al. Hypernetwork model and its properties[J]. Journal of Frontiers of Computer Science and Technology, 2017, 11(2): 194-211.
刘胜久, 李天瑞, 洪西进, 等. 超网络模型构建及特性分析[J]. 计算机科学与探索, 2017, 11(2): 194-211.
[5] LIU S J, LI T R, YANG Z L, et al. Measure and properties of weighted hypernetwork[J]. Journal of Computer Applica-tions, 2019, 39(11): 3107-3113.
刘胜久, 李天瑞, 杨宗霖, 等. 带权超网络的度量方法及其性质[J]. 计算机应用, 2019, 39(11): 3107-3113.
[6] LIU S J, LI T R, LIU J, et al. Research on multi-fractals of weighted hypernetwork[J]. Journal of University of Science and Technology of China, 2020, 50(3): 369-381.
刘胜久, 李天瑞, 刘佳, 等. 带权超网络的多重分形研究[J]. 中国科学技术大学学报, 2020, 50(3): 369-381.
[7] GUTMAN I. The energy of a graph[J]. Berichte der Mathe-matisch-Statistischen Sektion im Forschungszentrum Graz, 1978, 103: 1-22.
[8] ADIGA C, BALAKRISHNAN R, SO W. The skew energy of a digraph[J]. Linear Algebra and Its Applications, 2010, 432: 1825-1835.
[9] LIU J X, LI X L. Hermitian-adjacency matrices and Hermi-tian energies of mixed graphs[J]. Linear Algebra and Its Applications, 2015, 466: 182-207.
[10] INDULAL G, GUTMAN I, VIJAYAKUMAR A. On distance energy of graphs[J]. MATCH Communications in Mathema-tical and in Computer Chemistry, 2008, 60: 461-472.
[11] LIU J P, LIU B L. A Laplacian-energy like invariant of a graph[J]. MATCH Communications in Mathematical and in Computer Chemistry, 2008, 59: 355-372.
[12] JOOYANDEH M, KIANI D, MIRZAKHAH M. Incidence energy of a graph[J]. MATCH Communications in Mathe-matical and in Computer Chemistry, 2009, 62: 561-572.
[13] GUTMAN I, WAGNER S. The matching energy of a graph[J]. Discrete Applied Mathematics, 2012, 160: 2177-2187.
[14] GUTMAN I, ZHOU B. Laplacian energy of a graph[J]. Linear Algebra and Its Applications, 2006, 414: 29-37.
[15] SO W, ROBBIANO M, DE ABREU N. Applications of a theorem by Ky Fan in the theory of graph energy[J]. Linear Algebra and Its Applications, 2010, 432: 2163-2169.
[16] BRYC W, DEMBO A, JIANG T. Spectral measure of large random Hankel, Markov and Toeplitz matrices[J]. The Annals of Probability, 2006, 34: 1-38.
[17] DASA K, AOUCHICHE M, HANSEN P. On (distance) Lapl-acian energy and (distance) signless Laplacian energy of gr-aphs[J]. Discrete Applied Mathematics, 2018, 243: 172-185.
[18] LIU S J, LI T R, LIU X W. Network dimension: a new mea-sure for complex networks[J]. Computer Science, 2019, 46(1): 51-56.
刘胜久, 李天瑞, 刘小伟. 网络维数: 一种度量复杂网络的新方法[J]. 计算机科学, 2019, 46(1): 51-56.
[19] LIU S J, LI T R, ZHU J, et al. Network energy: a new energy of a graph[C]//Proceedings of the 14th IEEE International Con-ference on Intelligent Systems and Knowledge Engineering, Dalian, Nov 14-16, 2019. Piscataway: IEEE, 2019: 785-789.
[20] LIU S J, LI T R, ZHANG X B, et al. On network energy of oriented graphs[C]//Proceedings of the 14th International FLINS Conference on Robotics and Artificial Intelligence/IEEE 15th International Conference on Intelligent Systems and Knowledge Engineering, Cologne, Sep 30-Oct 3, 2020. Singapore: World Scientific, 2020: 11-18.
[21] WANG W Z, ZHOU K Q. On the Hermitian-incidence energy of mixed graphs[J]. Journal of Shandong University (Natural Science), 2019, 54(6): 53-58.
王维忠, 周琨强. 混合图的埃尔米特-关联能量[J]. 山东大学学报(理学版), 2019, 54(6): 53-58.
[22] WANG J F. Theoretical principle of hypergraph[M]. Beijing: Higher Education Press, 2006.
王建方. 超图的理论基础[M]. 北京: 高等教育出版社, 2006.