scholarly journals The number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs

2013 ◽  
Vol 392 (12) ◽  
pp. 2803-2806 ◽  
Author(s):  
Francesc Comellas ◽  
Alícia Miralles ◽  
Hongxiao Liu ◽  
Zhongzhi Zhang
Keyword(s):  
Spanning Trees ◽  
Infinite Family ◽  
Small World ◽  
Self Similar ◽  
Number Of Spanning Trees

The Number of Spanning Trees in Self-Similar Graphs

2011 ◽  
Vol 15 (2) ◽  
pp. 355-380 ◽  
Author(s):  
Elmar Teufl ◽  
Stephan Wagner
Keyword(s):  
Spanning Trees ◽  
Self Similar ◽  
Number Of Spanning Trees

scholarly journals Spanning trees of finite Sierpiński graphs

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Elmar Teufl ◽  
Stephan Wagner
Keyword(s):  
Spanning Trees ◽  
Dynamical Behavior ◽  
The Self ◽  
3 Dimensional ◽  
Sierpiński Graphs ◽  
Polynomial Map ◽  
International Audience ◽  
Self Similar ◽  
Number Of Spanning Trees

International audience We show that the number of spanning trees in the finite Sierpiński graph of level $n$ is given by $\sqrt[4]{\frac{3}{20}} (\frac{5}{3})^{-n/2} (\sqrt[4]{540})^{3^n}$. The proof proceeds in two steps: First, we show that the number of spanning trees and two further quantities satisfy a $3$-dimensional polynomial recursion using the self-similar structure. Secondly, it turns out, that the dynamical behavior of the recursion is given by a $2$-dimensional polynomial map, whose iterates can be computed explicitly.

scholarly journals The Evaluation of the Number and the Entropy of Spanning Trees on Generalized Small-World Networks

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Raihana Mokhlissi ◽  
Dounia Lotfi ◽  
Joyati Debnath ◽  
Mohamed El Marraki ◽  
Noussaima EL Khattabi
Keyword(s):  
Complex Networks ◽  
Spanning Tree ◽  
Decomposition Method ◽  
Spanning Trees ◽  
Small World ◽  
Theoretical Computer Science ◽  
The Other ◽  
Small World Networks ◽  
Theoretical Computer ◽  
Number Of Spanning Trees

Spanning trees have been widely investigated in many aspects of mathematics: theoretical computer science, combinatorics, so on. An important issue is to compute the number of these spanning trees. This number remains a challenge, particularly for large and complex networks. As a model of complex networks, we study two families of generalized small-world networks, namely, the Small-World Exponential and the Koch networks, by changing the size and the dimension of the cyclic subgraphs. We introduce their construction and their structural properties which are built in an iterative way. We propose a decomposition method for counting their number of spanning trees and we obtain the exact formulas, which are then verified by numerical simulations. From this number, we find their spanning tree entropy, which is lower than that of the other networks having the same average degree. This entropy allows quantifying the robustness of the networks and characterizing their structures.

Study on Applications of Laplacian Spectra for a Network

2013 ◽  
Vol 753-755 ◽  
pp. 2859-2862
Author(s):  
Hai Tang Wang
Keyword(s):  
Recurrence Relations ◽  
Spanning Trees ◽  
Small World ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Scale Free ◽  
Structure And Dynamics ◽  
Laplacian Spectra ◽  
Number Of Spanning Trees ◽  
Biological Field

Systems composing of dynamical units are ubiquitous in nature, ranging from physical to technological, and to biological field. These systems can be naturally described by networks, knowledge of its Laplacian eigenvalues is central to understanding its structure and dynamics for a network. In this paper, we study the Laplacian spectra of a family with scale-free and small-world properties. Based on the obtained recurrence relations, we determine explicitly the product of all nonzero Laplacian eigenvalues, as well as the sum of the reciprocals of these eigenvalues. Then, using these results, we further evaluate the number of spanning trees, Kirchhoff index.

scholarly journals Asymptotic enumeration on self-similar graphs with two boundary vertices

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Elmar Teufl ◽  
Stephan Wagner
Keyword(s):  
Spanning Trees ◽  
General Setting ◽  
Scaling Factor ◽  
Asymptotic Enumeration ◽  
Spanning Forests ◽  
International Audience ◽  
Self Similar ◽  
Connected Subgraphs ◽  
Graph Parameters ◽  
Number Of Spanning Trees

Combinatorics International audience We study two graph parameters, namely the number of spanning forests and the number of connected subgraphs, for self-similar graphs with exactly two boundary vertices. In both cases, we determine the general behavior for these and related auxiliary quantities by means of polynomial recurrences and a careful asymptotic analysis. It turns out that the so-called resistance scaling factor of a graph plays an essential role in both instances, a phenomenon that was previously observed for the number of spanning trees. Several explicit examples show that our findings are likely to hold in an even more general setting.

Sign in / Sign up

Export Citation Format

Share Document