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: [email protected]
Homepage: http://homepage.fudan.edu.cn/~zhangzz/
Blog: http://group.sciencenet.cn/home.php?mod=space&uid=311410
Main contents
1
Introduction to enumeration problems
 Spanning trees: theory and applications
 Matching (monomer and dimer)
 Perfect matching (dimers)
2
Our works
 Spanning trees on networks
 Matching and perfect matching on scalefree networks
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Various enumeration problems
 Spanning trees
 Matchings
 Perfect matching
 Spanning forests
 Spanning connected subgraphs
 Independent sets
 Acyclic orientations
 •••••••
3/45
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Definition of spanning tree
 A spanning tree of any connected network is
defined as a minimal set of edges that connect
every node.
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

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Relevance to sandpile model
 The number of spanning trees equals the number of
recurrent configurations.
The Electronic Journal
of Combinatorics.
2008,15, #R109.
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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 +
( u ,v )
N ST
(G )
R(u, v) 
N ST (G )
R(u,v)= C(u,v)/ (2m)
dz is degree of z, m is the number of edges
C(u,v) = F(s,t) + F(t,s) =2mR(u,v),
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Counting spanning trees
 Adjacency matrix A
 Diagonal degree matrix D
1 N
N ST (G )   i
N i 2
 Laplacian matrix L=D-A
 Probability transition matrix
 Normalized adjacency matrix
 Normalized Laplacian matrix
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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
 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
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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.
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Spanning trees in Apollonian networks
Confirm the conclusion on the last slide.
Journal of Mathematical Physics, 2011, 53: 113303
Submitted to Discrete Applied Mathmatics.
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Spanning trees in Koch networks
 Spanning trees
 Spanning forests
 Connected spanning subgraphs
Journal of Physics A, 2010, 43: 395102
Journal of Physics A, 2012, 45: 025102
13/43
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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.
14/43
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Spanning trees in fractal lattices: Spectral approach
 Kemeny constant
 Spanning trees
Chaos, 2012, 83:016116.(in press)
Spectra of transition matrix for Hanoi graphs
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The Hanoi towers game
What is the
minimum
number of
moves
?
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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.
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Spanning trees in small-world Farey graph
Farey sequence of order n denoted by
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Spanning trees in Farey graph
Theoretical Computer Science, 2011, 412:865–875
Two nodes and
they satisfy
are linked to each other if
Physica A, 2012, 391:3342-3349
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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.
2017/7/28
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.
2017/7/28
Thank You!