an:07033132
Zbl 1412.82011
Liao, Yunhua; Xie, Xiaoliang; Hou, Yaoping; Aziz-Alaoui, M. A.
Tutte polynomials of two self-similar network models
EN
J. Stat. Phys. 174, No. 4, 893-905 (2019).
00429302
2019
j
82B20 05C05 05C82 05C31
Tutte polynomial; number of spanning trees; complex network model; subdivision; inflation
Summary: 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.