Structural and spectral properties of corona graphs

Sharma, Rohan; Adhikari, Basudam; Mishra, Abhishek

doi:10.1016/j.dam.2017.01.005

“…The total number of vertices in G (k) are V (1 + V ) k . In [45], it is proved that the degree distribution of G (k) is a power law degree distribution, which fulfills the conditions for being real world network. It is also proved that the Laplacian matrix L(G), is a matrix of order (n 1 + n 1 n 2 ) whose eigenvalues are given by (a)…”

Section: Kλmentioning

confidence: 67%

From Sharma-Mittal to von-Neumann Entropy of a Graph

Mazumdar,

Singh,

Dutta

et al. 2019

Preprint

0

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In this article, we introduce the Sharma-Mittal entropy of a graph, which is a generalization of the existing idea of the von-Neumann entropy. The well-known Rényi, Thallis, and von-Neumann entropies can be expressed as limiting cases of Sharma-Mittal entropy. We have explicitly calculated them for cycle, path, and complete graphs. Also, we have proposed a number of bounds for these entropies. In addition, we have also discussed the entropy of product graphs, such as Cartesian, Kronecker, Lexicographic, Strong, and Corona products. The change in entropy can also be utilized in the analysis of growing network models (Corona graphs), useful in generating complex networks.

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“…In [14], the extremal edge-version ABC index of some graph operations was given. For more information, see [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

The Maximal ABC Index of the Corona of Two Graphs

Liu

¹

,

Shao

²

2021

Mathematical Problems in Engineering

0

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Let G 1 ∘ G 2 be the corona of two graphs G 1 and G 2 which is the graph obtained by taking one copy of G 1 and V G 1 copies of G 2 and then joining the i th vertex of G 1 to every vertex in the i th copy of G 2 . The atom-bond connectivity index (ABC index) of a graph G is defined as A B C G = ∑ u v ∈ E G d G u + d G v − 2 / d G u d G v , where E G is the edge set of G and d G u and d G v are degrees of vertices u and v , respectively. For the ABC indices of G 1 ∘ G 2 with G 1 and G 2 being connected graphs, we get the following results. (1) Let G 1 and G 2 be connected graphs. The ABC index of G 1 ∘ G 2 attains the maximum value if and only if both G 1 and G 2 are complete graphs. If the ABC index of G 1 ∘ G 2 attains the minimum value, then G 1 and G 2 must be trees. (2) Let T 1 and T 2 be trees. Then, the ABC index of T 1 ∘ T 2 attains the maximum value if and only if T 1 is a path and T 2 is a star.

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“…To this end, considerable deterministic network models that can reproduce common structural properties of real networks had been put forward to serve as test-bed [12]- [15]. Given the observation that massive networks are always composed of small pieces such as communities, modules and motifs [16]- [18], quite a few deterministic networks had been built out of smaller networks by employing graph products such as hierarchical product [19], [20], corona product [21], [22].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we study two types of random walks on a family of weighted heterogeneous networks induced by extended corona product [18], [21], [22] with different numbers of traps. In more detail, we introduce the weighted heterogeneous networks proposed by Qi et al in Ref [18] in section 2.…”

Section: Introductionmentioning

confidence: 99%

Exploring Trapping Problem on Weighted Heterogeneous Networks Induced by Extended Corona Product

Xu

¹

,

Wu

²

2020

IEEE Access

1

0

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Since many dynamical processes can be analyzed in the framework of trapping problem, it is great important to explore trapping problem on diverse complex systems or networks. However, addressing trapping problem on networks generated by graph products is still less touched. In this paper, by taking type of random walks and number of traps into account and compiling four different scenarios, trapping problem is touched more completely on resultant weighted heterogeneous networks of extended corona product and weight reinforcement mechanism. In more detail, we study standard random walk and delayed random walk on the network with a deep trap fixed at one initial node in the first and second scenarios respectively. This two types of random walks are further investigated on the network with three traps placed at initial three nodes in more challenging third and fourth scenarios. In all this four scenarios, the solutions of average trapping time(AT T) are deduced analytically to measure trapping efficiency, which agree well with their corresponding numerical counterparts and show that AT T grows sub-linearly with network size. Besides, expressions of AT T obtained in the second and fourth scenarios indicate that the parameter p governing delayed random walk alters the pre-factor of AT T but has no effect on the leading scaling of AT T. Furthermore, the comparisons between expression of AT T in first and third scenarios, expression of AT T in second and fourth scenarios imply that AT T can be lowered and trapping efficiency can be improved accordingly by introducing more traps. In summary, this work may enrich the clues for understanding trapping issue and modulating trapping process on more general heterogeneous weighted networks.

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Structural and spectral properties of corona graphs

Cited by 26 publications

References 38 publications

From Sharma-Mittal to von-Neumann Entropy of a Graph

From Sharma-Mittal to von-Neumann Entropy of a Graph

The Maximal ABC Index of the Corona of Two Graphs

Exploring Trapping Problem on Weighted Heterogeneous Networks Induced by Extended Corona Product

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