Tutte Polynomials of Two Self-similar Network Models
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The Tutte polynomial T(G; x, y) of a graph G, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in combinatorics and statistical physics. Graph operations have been extensively applied to model complex networks recently. In this paper, we study the Tutte polynomials of the diamond hierarchical lattices and a class of self-similar fractal models which can be constructed through graph operations. Firstly, we find out the behavior of the Tutte polynomial under k-inflation and k-subdivision which are two graph operations. Secondly, we compute and gain the Tutte polynomials of this two self-similar fractal models by using their structure characteristic. Moreover, as an application of the obtained results, some evaluations of their Tutte polynomials are derived, such as the number of spanning trees and the number of spanning forests.
KeywordsTutte polynomial The number of spanning trees Complex network model Subdivision Inflation
The authors thank the anonymous referees for their valuable comments and helpful suggestions, which have considerably improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (No. 11571101), the Scientific Research Fund of Hunan Provincial Education Department (No. 16C0872) and the Hunan Provincial Natural Science Foundation of China (No. 2018JJ3255). Yunhua Liao and M. A. Aziz-Alaoui were supported by Normandie region France and the XTerm ERDF project (European Regional Development Fund) on Complex Networks and Applications.