a r X i v :1008.0267v 1 [c o n d -m a t .s t a t -m e c h ] 2 A u g 2010epl draft
Introduction.–The enumeration of spanning trees in networks (graphs)is a fundamental issue in mathemat-ics [1–3],physics [4,5],and other discipline [6].A spanning tree of any connected network is defined as a minimal t of edges that connect every node.The problem of span-ning trees is relevant to various aspects of networks,such as reliability [7,8],optimal synchronization [9],standard random walks [10],and loop-erad random walks [11].In particular,the number of spanning trees corresponds to the partition function of the q -state Potts model [12]in the limit of q approaching zero,which in turn cloly relates to the sandpile model [13].Becau of the diver applications in a number of fields [14],a lot of efforts have been devoted to the study of spanning trees.For example,the exact number of span-ning trees in regular lattices [4,15]and Sierpinski gas-kets [5]has been explicitly determined in previous studies.However,regular lattices and fractals cannot well mimic the real-life networks,which have been recently found to synchronously exhibit two striking properties:scale-free behavior [16]and small-world effects [17]that has a strong impact on the enumeration problems on networks.For example,previous work on counting subgraphs,such as cliques [18],loops and Hamiltonian cycles [19],
has
5.Alternatively,the network can be also created in an-
other method[23,24].Given the generation n,G n+1may be obtained by joining at the hubs(the most connected nodes)three copies of G n,e Fig.2.According to the latter construction algorithm,we can easily compute the network order of G n is V n=3n+1+3
V n i=V n−1
i=1
λi(n),(1)
whereλi(n)(i=1,2,...,V n−1)are the V n−1nonzero eigenvalues of the Laplacian matrix for G n.For a network, the non-diagonal element l ij(i=j)of its Laplacian matrix is-1(or0)if nodes i and j are(or not)directly connected, while the diagonal entry l ii equals the degree of node i. Using Eq.(1),we can calculate directly the number of spanning trees N ST(n)of G n(e Fig.3).From Fig.3,we can e that
tayaN ST(n)approximately grows exponentially in V n.This allows to define the entropy of spanning trees for G n as the limiting value[1–3]
E G n=lim
V n→∞
香奈儿英文ln N ST(n)imperial college
a n
that obeys the following recursive relation
h n+1=t n+1
2a n
cityhall
=
3
2n
vansky.(6)
Then,
a n=
2n
short是什么意思
3n
(t n)3.(8)
Considering the initial value t0=3,we can solve Eq.(8) to obtain the explicit solution
possiblely
屏障是什么意思N ST(n)=t n=2(3n+1−2n−3)/43(3n+1+2n+1)/4.(9) Analogously,we can derive the exact formula for a n as:
a n=2(3n+1+2n−3)/43(3n+1−2n−3)/4.(10) It not difficult to reprent N ST(n)as a function of the network order V n,with the aim to obtain the relation be-tween the two quantities.Recalling V n=3n+1+3V n
siligo=枯肠
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