Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group

Ghorbani, Modjtaba; Hakimi-Nezhaad, Mardjan; Dehmer, Matthias; Li, Xueliang

doi:10.3390/sym12091411

Cited by 2 publications

(2 citation statements)

References 28 publications

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“…Creating polynomials on the set of orbits of a graph would help to distinguish vertices having different properties and thus to separate them into different orbits; see [19,20].…”

Section: Methods and Resultsmentioning

confidence: 99%

Network Analyzing by the Aid of Orbit Polynomial

Ghorbani

Dehmer

2021

Symmetry

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This article aims to be a further contribution to the research on structural complexity networks. Here, we emphasize measures to determine symmetry. The so-called “orbit polynomial” is defined by OG(x)=∑iaixi, where ai is the number of orbits of size i. Furthermore, the graph polynomial 1−OG(x) has a unique positive root in the interval (0,1), which can be considered as a relevant measure of the symmetry of a graph. In the present paper, we studied some properties of the orbit polynomial with respect to the stabilizer elements of each vertex. Furthermore, we constructed graphs with a small number of orbits and characterized some classes of graphs in terms of calculating their orbit polynomials. We studied the symmetry structure of well-known real-world networks in terms of the orbit polynomial.

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“…Creating polynomials on the set of orbits of a graph would help to distinguish vertices having different properties and thus to separate them into different orbits; see [19,20].…”

Section: Methods and Resultsmentioning

confidence: 99%

Network Analyzing by the Aid of Orbit Polynomial

Ghorbani

Dehmer

2021

Symmetry

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“…Automorphism groups help identify and classify symmetries in crystal lattices, aiding in the analysis and prediction of crystallographic patterns and properties. For example, reference (35,36) gives examples of the use of the apparatus of group theory in research on crystallography, quantum mechanics, and elementary particle physics. In particular, in these studies matrix groups and representations of unitary groups are actively used.…”

Section: Applications Of Automorphism Group In Real Lifementioning

confidence: 99%

Automorphism Group of Polyhedral Graphs

Ghorbani,

Alidehi-Ravandi,

Dehmer

2023

Preprint

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Symmetry, as a key concept in geometry and the foundational principle of mathematical concepts, is a fundamental part of geometry and nature, creating patterns that help us conceptually organize our world. It is often linked with aesthetic beauty and can be observed in both natural phenomena and human-made objects, such as artworks and manufactured products, across the globe. The automorphism group of a graph is a mathematical structure that captures all the symmetries of the graph. symmetry refers to the property of a graph, indicating that it can be transformed without changing its structure, while the automorphism group is the collection of all permutations of the vertices that preserve the graph's adjacency relationships. In graph theory, the concepts of symmetry group and automorphism group play fundamental roles in understanding the symmetries and structural properties of graphs. This paper aims to explore the relationship between symmetry groups and automorphism groups, shedding light on their interconnected nature and their significance in graph theory. We delve into the definitions and properties of both symmetry groups and automorphism groups, highlighting their similarities and distinctions. Furthermore, we discuss their applications in various fields, including computer science, chemistry, and network analysis. Through this exploration, we aim to provide a comprehensive understanding of the relationship between symmetry groups and automorphism groups, deepening our comprehension of the symmetries and structures inherent in graphs.

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Analysis of the Graovac–Pisanski Index of Some Polyhedral Graphs Based on Their Symmetry Group

Cited by 2 publications

References 28 publications

Network Analyzing by the Aid of Orbit Polynomial

Network Analyzing by the Aid of Orbit Polynomial

Automorphism Group of Polyhedral Graphs

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