Structural and spectral properties of a family of deterministic recursive trees: rigorous solutions

Qi, Yi; Zhang, Zhongzhi; Ding, Bailu; Zhou, Shuigeng; Guan, Jihong

doi:10.1088/1751-8113/42/16/165103

Cited by 8 publications

(7 citation statements)

References 74 publications

(111 reference statements)

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“…), the determination of the exact spectrum of a general graph is a difficult task. However, for some large families of trees, the recursivity of the graph construction helps to find a relationship between the characteristic polynomials at different iteration steps [22][23][24][25][26], which can be used to produce iteratively an analytical expression for the spectra. Here, we apply a similar technique for the generalized deterministic recursive trees (GDRT) introduced in [24] and known as r-adic hypertrees.…”

Section: Laplacian Spectra Of a Graph And Mean First Passage Timementioning

confidence: 99%

Mean first-passage time for random walks on generalized deterministic recursive trees

Comellas

Miralles

2010

Phys. Rev. E

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We describe a technique that allows the exact analytical computation of the mean first passage time (MFPT) for infinite families of trees using their recursive properties. The method is based in the relationship between the MFPT and the eigenvalues of the Laplacian matrix of the trees but avoids their explicit computation. We apply this technique to find the MFPT for a family of generalized deterministic recursive trees. The method, however, can be adapted to other self-similar tree families.

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Section: Laplacian Spectra Of a Graph And Mean First Passage Timementioning

confidence: 99%

Mean first-passage time for random walks on generalized deterministic recursive trees

Comellas

Miralles

2010

Phys. Rev. E

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“…We study this model because of its intrinsic inter-est [44][45][46][47][48] and its relevance to real-life networks. For instance, it is small-world [36,44,46,47,49]; particularly, the so-called border tree motifs have been shown to be present, in a significant way, in real-world systems [50].…”

Section: The Deterministic Uniform Recursive Treesmentioning

confidence: 99%

Explicit determination of mean first-passage time for random walks on deterministic uniform recursive trees

Zhang

Zhou

et al. 2010

Phys. Rev. E

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The determination of mean first-passage time (MFPT) for random walks in networks is a theoretical challenge, and is a topic of considerable recent interest within the physics community. In this paper, according to the known connections between MFPT, effective resistance, and the eigenvalues of graph Laplacian, we first study analytically the MFPT between all node pairs of a class of growing treelike networks, which we term deterministic uniform recursive trees (DURTs), since one of its particular cases is a deterministic version of the famous uniform recursive tree. The interesting quantity is determined exactly through the recursive relation of the Laplacian spectra obtained from the special construction of DURTs. The analytical result shows that the MFPT between all couples of nodes in DURTs varies as N ln N for large networks with node number N. Second, we study trapping on a particular network of DURTs, focusing on a special case with the immobile trap positioned at a node having largest degree. We determine exactly the average trapping time (ATT) that is defined as the average of FPT from all nodes to the trap. In contrast to the scaling of the MFPT, the leading behavior of ATT is a linear function of N. Interestingly, we show that the behavior for ATT of the trapping problem is related to the trapping location, which is in comparison with the phenomenon of trapping on fractal T-graph although both networks exhibit tree structure. Finally, we believe that the methods could open the way to exactly calculate the MFPT and ATT in a wide range of deterministic media.

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“…The betweenness distribution featuring power-law was also observed in [23], [24], [25]. In [20], the authors analysed a family of deterministic recursive trees having exponential decay in cumulative degree distribution as well as cumulative power-law betweenness distribution.…”

Section: Betweenness Distributionmentioning

confidence: 96%

“…Now, for assessing the number of shortest paths, we use reasoning similar to [20], [21]. Let all the nodes attached to node i be termed as offsprings.…”

Section: Betweenness Distributionmentioning

confidence: 99%

Structural and spectral properties of corona graphs

Sharma

Adhikari

Mishra

2017

Discrete Applied Mathematics

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Product graphs have been gainfully used in literature to generate mathematical models of complex networks which inherit properties of real networks. Realizing the duplication phenomena imbibed in the definition of corona product of two graphs, we define Corona graphs. Given a small simple connected graph which we call seed graph, Corona graphs are defined by taking corona product of a seed graph iteratively. We show that the cumulative degree distribution of Corona graphs decay exponentially when the seed graph is regular and cumulative betweenness distribution follows power law when seed graph is a clique. We determine explicit formulae of eigenvalues, Laplacian eigenvalues and signless Laplacian eigenvalues of Corona graphs when the seed graph is regular. Computable expressions of eigenvalues and signless Laplacian eigenvalues of Corona graphs are also obtained when the seed graph is a star graph.

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Structural and spectral properties of a family of deterministic recursive trees: rigorous solutions

Cited by 8 publications

References 74 publications

Mean first-passage time for random walks on generalized deterministic recursive trees

Mean first-passage time for random walks on generalized deterministic recursive trees

Explicit determination of mean first-passage time for random walks on deterministic uniform recursive trees

Structural and spectral properties of corona graphs

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