Enumeration problems of networks 2012网络传播动力学研讨会 Enumeration problems of networks 章 忠 志 复旦大学计算机科学技术学院 Email: zhangzz@fudan.edu.cn Homepage: http://homepage.fudan.edu.cn/~zhangzz/ Blog: http://group.sciencenet.cn/home.php?mod=space&uid=311410

Main contents Introduction to enumeration problems Our works 1 Spanning trees: theory and applications Matching (monomer and dimer) Perfect matching (dimers) Our works 2 Spanning trees on networks Matching and perfect matching on scale-free networks 2019/2/16

Various enumeration problems Spanning trees Matchings Perfect matching Spanning forests Spanning connected subgraphs Independent sets Acyclic orientations ••••••• 2019/2/16

Definition of spanning tree A spanning tree of any connected network is defined as a minimal set of edges that connect every node. 图2-26给出的是一个包含5个顶点的连通图的4个生成树。

Applications and relevance of spanning trees A measure of reliability Loop-erased random walks q-state Potts model Sandpile model Electrical networks Isotropic random walks 2019/2/16

Relevance to sandpile model The number of spanning trees equals the number of recurrent configurations. The Electronic Journal of Combinatorics. 2008,15, #R109. 2019/2/16

Connections with electrical networks Every edge – a resistor of 1 ohm. Voltage difference of 1 volt between u and v. R(u,v) – inverse of electrical current from u to v. v _ + u C(u,v) = F(s,t) + F(t,s) =2mR(u,v), R(u,v)= C(u,v)/ (2m) dz is degree of z, m is the number of edges 2019/2/16

Counting spanning trees Adjacency matrix A Diagonal degree matrix D Laplacian matrix L=D-A Probability transition matrix Normalized adjacency matrix Normalized Laplacian matrix 2019/2/16

Definition of matching Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that is, no two edges share a common vertex. Maximal matching Maximum matching 图2-26给出的是一个包含5个顶点的连通图的4个生成树。 Perfect matching

Our works Enumerating spanning in various networks Scale-free networks: Pseudofractal scale-free web, Apollonian networks, Koch networks Fractal networks: Scale-free lattice, Hanoi graphs Small-world network Counting matching on scale-free networks Matching Perfect matching 2019/2/16

Spanning trees in pseudofractal scale-free web A counterintuitive conclusion that a network with more spanning trees may be relatively unreliable. EPL, 2010, 90:68002. 2019/2/16

Spanning trees in Apollonian networks Confirm the conclusion on the last slide. Journal of Mathematical Physics, 2011, 53: 113303 Submitted to Discrete Applied Mathmatics. 2019/2/16

Spanning trees in Koch networks Spanning forests Connected spanning subgraphs Journal of Physics A, 2010, 43: 395102 Journal of Physics A, 2012, 45: 025102 2019/2/16

Spanning trees in fractal scale-free lattices Fractality can significantly increase the number of spanning trees in fractal scale-free networks. Fractal dimension has a predominant influence on the number of spanning trees. Journal of Mathematical Physics, 2011, 53: 113303 Physical Review E, 2011, 83:016116. 2019/2/16

Spanning trees in fractal lattices: Spectral approach Kemeny constant Spanning trees Chaos, 2012, 83:016116.(in press)

Spectra of transition matrix for Hanoi graphs 2019/2/16

What is the minimum number of moves ? The Hanoi towers game   What is the minimum number of moves ? 2019/2/16

Spectra of Hanoi graphs and applications Structural properties Spectral prosperities We obtain all the eigenvalues and their corresponding degeneracies. Spanning trees We determine the exact number of spanning trees and derive an explicit formula of the eigentime identity. Journal of Physics A, 2012, 45:345101. 2019/2/16

Spanning trees in small-world Farey graph Farey sequence of order n denoted by 2019/2/16

Spanning trees in Farey graph Theoretical Computer Science, 2011, 412:865–875 Two nodes and are linked to each other if they satisfy Physica A, 2012, 391:3342-3349 2019/2/16

Monomer-dimer in pseudofractal scale-free web We obtain the exact formula for the number of all possible monomer–dimer arrangements on the network. Physica A, 2012, 391: 828–833. 2019/2/16

Perfect matching in scale-free networks Non-fractal scale-free network Fractal scale-free network We obtain the explicit expression of the number of perfect matching of the two scale-free networks. Submitted to Theoretical Computer Science. 2019/2/16

Thank You!