Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients

Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients

Suppose <i>a<sub>i</sub></i> indicates the number of orbits of size <i>i</i> in graph <i>G</i>. A new counting polynomial, namely an orbit polynomial, is defined as <i>O<sub>G</sub></i>(<i>x</i>) = ∑<i><sub&...

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Main Authors: Modjtaba Ghorbani, Maryam Jalali-Rad, Matthias Dehmer
Format: Article
Language:English
Published: MDPI AG 2021-04-01
Series:Symmetry
Subjects:
orbit
group action
polynomial roots
orbit-stabilizer theorem
Online Access:https://www.mdpi.com/2073-8994/13/4/710
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spelling doaj-6b3abdd6559f4c0fa1557ee7c95debaf2021-04-17T23:03:42ZengMDPI AGSymmetry2073-89942021-04-011371071010.3390/sym13040710Orbit Polynomial of Graphs Versus Polynomial with Integer CoefficientsModjtaba Ghorbani0Maryam Jalali-Rad1Matthias Dehmer2Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-136, IranDepartment of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-136, IranDepartment of Computer Science, Swiss Distance University of Applied Sciences, 3900 Brig, SwitzerlandSuppose <i>a<sub>i</sub></i> indicates the number of orbits of size <i>i</i> in graph <i>G</i>. A new counting polynomial, namely an orbit polynomial, is defined as <i>O<sub>G</sub></i>(<i>x</i>) = ∑<i><sub>i</sub> a<sub>i</sub>x<sup>i</sup></i>. Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials.https://www.mdpi.com/2073-8994/13/4/710orbitgroup actionpolynomial rootsorbit-stabilizer theorem
collection DOAJ
language English
format Article
sources DOAJ
author Modjtaba Ghorbani
Maryam Jalali-Rad
Matthias Dehmer
spellingShingle Modjtaba Ghorbani
Maryam Jalali-Rad
Matthias Dehmer
Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients
Symmetry
orbit
group action
polynomial roots
orbit-stabilizer theorem
author_facet Modjtaba Ghorbani
Maryam Jalali-Rad
Matthias Dehmer
author_sort Modjtaba Ghorbani
title Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients
title_short Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients
title_full Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients
title_fullStr Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients
title_full_unstemmed Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients
title_sort orbit polynomial of graphs versus polynomial with integer coefficients
publisher MDPI AG
series Symmetry
issn 2073-8994
publishDate 2021-04-01
description Suppose <i>a<sub>i</sub></i> indicates the number of orbits of size <i>i</i> in graph <i>G</i>. A new counting polynomial, namely an orbit polynomial, is defined as <i>O<sub>G</sub></i>(<i>x</i>) = ∑<i><sub>i</sub> a<sub>i</sub>x<sup>i</sup></i>. Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials.
topic orbit
group action
polynomial roots
orbit-stabilizer theorem
url https://www.mdpi.com/2073-8994/13/4/710
work_keys_str_mv AT modjtabaghorbani orbitpolynomialofgraphsversuspolynomialwithintegercoefficients
AT maryamjalalirad orbitpolynomialofgraphsversuspolynomialwithintegercoefficients
AT matthiasdehmer orbitpolynomialofgraphsversuspolynomialwithintegercoefficients
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