Hamming dimension of a graph—The case of Sierpiński graphs

Klavar, Sandi; Peterin, Iztok; Zemljič, Sara Sabrina

doi:10.1016/j.ejc.2012.09.006

Cited by 19 publications

(10 citation statements)

References 22 publications

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“…Sierpinski gasket graphs S n [28] are associated with the Sierpinski gasket -well-known topological fractal with a Hausdorff dimension log(3)/ log(2) ≈ 1.585. Edges of S n are line segments of the n-th approximation of the Sierpinski gasket, and vertices are intersection points of these segments (Fig.…”

Section: Sierpinski Gasket Graphsmentioning

confidence: 99%

“…Here K = ∆(G) + 1 is an upper bound on the Hausdorff dimension of the graph G. The binary variable x i,k indicates whether the clique C i is colored by a color k, and the binary variable y k indicates whether the color k is used; the relation between these variables is enforced by the constraints (26). The constraints (28) state that every clique receives at most one color; it is possible that a clique does not have any color, which means that a clique is not selected as a cluster. By the constraints (28), all cliques containing any given vertex v receive different colors, and the constraints (29) ensure that at least one of the cliques covering any edge uv receives a color (i.e.…”

Section: Calculation Of Lebesgue and Hausdorff Dimensionsmentioning

confidence: 99%

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Graph fractal dimension and structure of fractal networks: a combinatorial perspective

Skums,

Bunimovich

2019

Preprint

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In this paper we study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product (or Prague or Nešetřil-Rödl) dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colorings and graph Kolmogorov complexity, and analyze the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework to evolutionary studies by revealing the growth of self-organization of heterogeneous viral populations over the course of their intra-host evolution, thus suggesting mechanisms of their gradual adaptation to the host's environment. As far as the authors know, the proposed approach is the first theoretical framework for study of network fractality within the combinatorial paradigm. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

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Section: Sierpinski Gasket Graphsmentioning

confidence: 99%

Section: Calculation Of Lebesgue and Hausdorff Dimensionsmentioning

confidence: 99%

Graph fractal dimension and structure of fractal networks: a combinatorial perspective

Skums,

Bunimovich

2019

Preprint

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“…Later, those graphs have been called Sierpiński graphs in [7] and they were studied by now from numerous points of view. The reader is invited to read, for instance, the following recent papers [2,5,4,7,8,9] and references therein. This construction was generalized in [3] for any graph G, by defining the t-th generalized Sierpiński graph of G, denoted by S(G, t), as the graph with vertex set V t (G) and edge set defined as follows.…”

Section: Preliminaries On Generalized Sierpiński Graphsmentioning

confidence: 99%

The strong metric dimension of generalized Sierpiński graphs with pendant vertices

Estaji

Rodríguez-Velázquez

2016

Ars Math. Contemp.

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“…In those publications, geometric objects were considered as one of the types of fractals [33]. Later, in the 2000s, publications appeared offering definitions of Sierpinski graphs from the classical point of view, when a graph is given sets of vertices and edges [34][35][36]. Recently, researchers have proposed a uniform definition of Sierpinski graphs [37,38].…”

Section: Introductionmentioning

confidence: 99%

Research of NP-Complete Problems in the Class of Prefractal Graphs

Kochkarov

2021

Mathematics

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NP-complete problems in graphs, such as enumeration and the selection of subgraphs with given characteristics, become especially relevant for large graphs and networks. Herein, particular statements with constraints are proposed to solve such problems, and subclasses of graphs are distinguished. We propose a class of prefractal graphs and review particular statements of NP-complete problems. As an example, algorithms for searching for spanning trees and packing bipartite graphs are proposed. The developed algorithms are polynomial and based on well-known algorithms and are used in the form of procedures. We propose to use the class of prefractal graphs as a tool for studying NP-complete problems and identifying conditions for their solvability. Using prefractal graphs for the modeling of large graphs and networks, it is possible to obtain approximate solutions, and some exact solutions, for problems on natural objects—social networks, transport networks, etc.

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Hamming dimension of a graph—The case of Sierpiński graphs

Cited by 19 publications

References 22 publications

Graph fractal dimension and structure of fractal networks: a combinatorial perspective

Graph fractal dimension and structure of fractal networks: a combinatorial perspective

The strong metric dimension of generalized Sierpiński graphs with pendant vertices

Research of NP-Complete Problems in the Class of Prefractal Graphs

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