scholarly journals On distances in Sierpiński graphs: Almost-extreme vertices and metric dimension

2013 ◽  
Vol 7 (1) ◽  
pp. 72-82 ◽  
Author(s):  
Sandi Klavzar ◽  
Sara Zemljic
Keyword(s):  
Computer Science ◽  
Metric Dimension ◽  
Fractal Nature ◽  
Explicit Formulas ◽  
Arbitrary Vertex ◽  
Tower Of Hanoi ◽  
Sierpiński Graphs ◽  
Total Distance

Sierpi?ski graphs Sn p form an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. An almost-extreme vertex of Sn p is introduced as a vertex that is either adjacent to an extreme vertex of Sn p or is incident to an edge between two subgraphs of Sn p isomorphic to Snp-1. Explicit formulas are given for the distance in Sn p between an arbitrary vertex and an almostextreme vertex. The formulas are applied to compute the total distance of almost-extreme vertices and to obtain the metric dimension of Sierpi?ski graphs.

scholarly journals Some topological properties of uniform subdivision of Sierpiński graphs

2021 ◽  
Vol 44 (1) ◽  
pp. 218-227
Author(s):  
Jia-Bao Liu ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Muhammad Faisal Nadeem ◽  
Muhammad Ahsan Binyamin
Keyword(s):  
Topological Index ◽  
Topological Properties ◽  
Fractal Nature ◽  
Tower Of Hanoi ◽  
Zagreb Indices ◽  
Chemical Structures ◽  
Sierpiński Graphs ◽  
The Family ◽  
Forgotten Index ◽  
Basic Graph

Abstract Sierpiński graphs are family of fractal nature graphs having applications in mathematics of Tower of Hanoi, topology, computer science, and many more diverse areas of science and technology. This family of graphs can be generated by taking certain number of copies of the same basic graph. A topological index is the number which shows some basic properties of the chemical structures. This article deals with degree based topological indices of uniform subdivision of the generalized Sierpiński graphs S(n,G) and Sierpiński gasket Sn . The closed formulae for the computation of different kinds of Zagreb indices, multiple Zagreb indices, reduced Zagreb indices, augmented Zagreb indices, Narumi-Katayama index, forgotten index, and Zagreb polynomials have been presented for the family of graphs.

scholarly journals Polynomials and General Degree-Based Topological Indices of Generalized Sierpinski Networks

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Chengmei Fan ◽  
M. Mobeen Munir ◽  
Zafar Hussain ◽  
Muhammad Athar ◽  
Jia-Bao Liu
Keyword(s):  
Complex Systems ◽  
Computer Science ◽  
Topological Indices ◽  
Fractal Nature ◽  
Similar Nature ◽  
Zagreb Index ◽  
Recursive Sequences ◽  
And Mathematics

Sierpinski networks are networks of fractal nature having several applications in computer science, music, chemistry, and mathematics. These networks are commonly used in chaos, fractals, recursive sequences, and complex systems. In this article, we compute various connectivity polynomials such as M -polynomial, Zagreb polynomials, and forgotten polynomial of generalized Sierpinski networks S k n and recover some well-known degree-based topological indices from these. We also compute the most general Zagreb index known as α , β -Zagreb index and several other general indices of similar nature for this network. Our results are the natural generalizations of already available results for particular classes of such type of networks.

scholarly journals On Three Constructions of Nanotori

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2036
Author(s):  
Vesna Andova ◽  
Pavel Dimovski ◽  
Martin Knor ◽  
Riste Škrekovski
Keyword(s):  
Computer Science ◽  
Explicit Form ◽  
Topological Indices ◽  
The Other ◽  
Explicit Formulas ◽  
Extra Parameter

There are three different approaches for constructing nanotori in the literature: one with three parameters suggested by Altshuler, another with four parameters used mostly in chemistry and physics after the discovery of fullerene molecules, and one with three parameters used in interconnecting networks of computer science known under the name generalized honeycomb tori. Altshuler showed that his method gives all non-isomorphic nanotori, but this was not known for the other two constructions. Here, we show that these three approaches are equivalent and give explicit formulas that convert parameters of one construction into the parameters of the other two constructions. As a consequence, we obtain that the other two approaches also construct all nanotori. The four parameters construction is mainly used in chemistry and physics to describe carbon nanotori molecules. Some properties of the nanotori can be predicted from these four parameters. We characterize when two different quadruples define isomorphic nanotori. Even more, we give an explicit form of all isomorphic nanotori (defined with four parameters). As a consequence, infinitely many 4-tuples correspond to each nanotorus; this is due to redundancy of having an “extra” parameter, which is not a case with the other two constructions. This result significantly narrows the realm of search of the molecule with desired properties. The equivalence of these three constructions can be used for evaluating different graph measures as topological indices, etc.

scholarly journals On distances in generalized Sierpiński graphs

2018 ◽  
Vol 12 (1) ◽  
pp. 49-69 ◽  
Author(s):  
Alejandro Estrada-Moreno ◽  
Erick Rodríguez-Bazan ◽  
Juan Rodríguez-Velázquez
Keyword(s):  
Explicit Formula ◽  
Recursive Formula ◽  
Arbitrary Vertex ◽  
Base Graph ◽  
Sierpiński Graphs ◽  
Recursive Formulas

In this paper we propose formulas for the distance between vertices of a generalized Sierpi?ski graph S(G, t) in terms of the distance between vertices of the base graph G. In particular, we deduce a recursive formula for the distance between an arbitrary vertex and an extreme vertex of S(G, t), and we obtain a recursive formula for the distance between two arbitrary vertices of S(G, t) when the base graph is triangle-free. From these recursive formulas, we provide algorithms to compute the distance between vertices of S(G, t). In addition, we give an explicit formula for the diameter and radius of S(G, t) when the base graph is a tree.

scholarly journals The strong metric dimension of generalized Sierpiński graphs with pendant vertices

2016 ◽  
Vol 12 (1) ◽  
pp. 127-134 ◽  
Author(s):  
Ehsan Estaji ◽  
Juan Alberto Rodríguez-Velázquez
Keyword(s):  
Metric Dimension ◽  
Sierpiński Graphs ◽  
Strong Metric Dimension

scholarly journals On the Upper Bounds of Fractional Metric Dimension of Symmetric Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Muhammad Javaid ◽  
Muhammad Kamran Aslam ◽  
Jia-Bao Liu
Keyword(s):  
Image Processing ◽  
Drug Discovery ◽  
Computer Science ◽  
Chemical Compounds ◽  
Upper Bounds ◽  
Metric Dimension ◽  
Rotationally Symmetric ◽  
Metric Dimensions ◽  
Structural Aspects ◽  
Symmetric Networks

Distance-based numeric parameters play a pivotal role in studying the structural aspects of networks which include connectivity, accessibility, centrality, clustering modularity, complexity, vulnerability, and robustness. Several tools like these also help to resolve the issues faced by the different branches of computer science and chemistry, namely, navigation, image processing, biometry, drug discovery, and similarities in chemical compounds. For this purpose, in this article, we are considering a family of networks that exhibits rotationally symmetric behaviour known as circular ladders consisting of triangular, quadrangular, and pentagonal faced ladders. We evaluate their upper bounds of fractional metric dimensions of the aforementioned networks.

scholarly journals Sensitivities and block sensitivities of elementary symmetric Boolean functions

2021 ◽  
Vol 15 (1) ◽  
pp. 434-453
Author(s):  
Jing Zhang ◽  
Yuan Li ◽  
John O. Adeyeye
Keyword(s):  
Information Technology ◽  
Computer Science ◽  
Boolean Functions ◽  
Regulatory Networks ◽  
Explicit Formulas ◽  
Complexity Measures ◽  
Symmetric Boolean Functions ◽  
Block Sensitivity

Abstract Boolean functions have important applications in molecular regulatory networks, engineering, cryptography, information technology, and computer science. Symmetric Boolean functions have received a lot of attention in several decades. Sensitivity and block sensitivity are important complexity measures of Boolean functions. In this paper, we study the sensitivity of elementary symmetric Boolean functions and obtain many explicit formulas. We also obtain a formula for the block sensitivity of symmetric Boolean functions and discuss its applications in elementary symmetric Boolean functions.

scholarly journals Sierpiński graphs as spanning subgraphs of Hanoi graphs

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Andreas Hinz ◽  
Sandi Klavžar ◽  
Sara Zemljič
Keyword(s):  
Topological Spaces ◽  
Tower Of Hanoi ◽  
Sierpiński Graphs ◽  
Spanning Subgraphs ◽  
Spanning Subgraph

AbstractHanoi graphs H pn model the Tower of Hanoi game with p pegs and n discs. Sierpinski graphs S pn arose in investigations of universal topological spaces and have meanwhile been studied extensively. It is proved that S pn embeds as a spanning subgraph into H pn if and only if p is odd or, trivially, if n = 1.

scholarly journals On Fault-Tolerant Resolving Sets of Some Families of Ladder Networks

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Hua Wang ◽  
Muhammad Azeem ◽  
Muhammad Faisal Nadeem ◽  
Ata Ur-Rehman ◽  
Adnan Aslam
Keyword(s):  
Computer Networks ◽  
Computer Network ◽  
Fault Tolerant ◽  
Metric Dimension ◽  
Resolving Set ◽  
Arbitrary Vertex

In computer networks, vertices represent hosts or servers, and edges represent as the connecting medium between them. In localization, some special vertices (resolving sets) are selected to locate the position of all vertices in a computer network. If an arbitrary vertex stopped working and selected vertices still remain the resolving set, then the chosen set is called as the fault-tolerant resolving set. The least number of vertices in such resolving sets is called the fault-tolerant metric dimension of the network. Because of the variety of applications of the metric dimension in different areas of sciences, many generalizations were proposed, and fault tolerant is one of them. In this paper, we computed the fault-tolerant metric dimension of triangular snake, ladder, Mobius ladder, and hexagonal ladder networks. It is important to observe that, in all these classes of networks, the fault-tolerant metric dimension and metric dimension differ by 1.

Sign in / Sign up

Export Citation Format

Share Document