On the Degeneracy of the Orbit Polynomial and Related Graph Polynomials

The orbit polynomial is a new graph counting polynomial which is defined as OG(x)=∑i=1rx|Oi|, where O1, …, Or are all vertex orbits of the graph G. In this article, we investigate the structural properties of the automorphism group of a graph by using several novel counting polynomials. Besides, we explore the orbit polynomial of a graph operation. Indeed, we compare the degeneracy of the orbit polynomial with a new graph polynomial based on both eigenvalues of a graph and the size of orbits.

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Graph polynomials and symmetries

Journal of Algebra and Its Applications ◽

10.1142/s021949881950172x ◽

2019 ◽

Vol 18 (09) ◽

pp. 1950172 ◽

Cited By ~ 1

Author(s):

Nafaa Chbili

Keyword(s):

Automorphism Group ◽

Finite Field ◽

Characteristic Polynomial ◽

Prime Order ◽

Tutte Polynomial ◽

Quotient Graph ◽

Graph Polynomials ◽

Tutte Polynomials

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

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ON THE KAUFFMAN–VOGEL AND THE MURAKAMI–OHTSUKI–YAMADA GRAPH POLYNOMIALS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216512500988 ◽

2012 ◽

Vol 21 (10) ◽

pp. 1250098 ◽

Cited By ~ 2

Author(s):

HAO WU

Keyword(s):

Simple Circuit ◽

Graph Polynomials ◽

Graph Polynomial ◽

Link Homology ◽

Kauffman Polynomial ◽

Colored Link

This paper consists of three parts. First, we generalize the Jaeger Formula to express the Kauffman–Vogel graph polynomial as a state sum of the Murakami–Ohtsuki–Yamada graph polynomial. Then, we demonstrate that reversing the orientation and the color of a MOY graph along a simple circuit does not change the 𝔰𝔩(N) Murakami–Ohtsuki–Yamada polynomial or the 𝔰𝔩(N) homology of this MOY graph. In fact, reversing the orientation and the color of a component of a colored link only changes the 𝔰𝔩(N) homology by an overall grading shift. Finally, as an application of the first two parts, we prove that the 𝔰𝔬(6) Kauffman polynomial is equal to the 2-colored 𝔰𝔩(4) Reshetikhin–Turaev link polynomial, which implies that the 2-colored 𝔰𝔩(4) link homology categorifies the 𝔰𝔬(6) Kauffman polynomial.

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A NOTE ON THE TUTTE POLYNOMIAL AND THE AUTOMORPHISM GROUP OF A GRAPH

Asian-European Journal of Mathematics ◽

10.1142/s1793557114500016 ◽

2014 ◽

Vol 07 (01) ◽

pp. 1450001 ◽

Cited By ~ 2

Author(s):

Nafaa Chbili

Keyword(s):

Automorphism Group ◽

Short Note ◽

Tutte Polynomial ◽

Necessary Condition ◽

Graph Polynomials

A graph G is said to be p-periodic if the automorphism group Aut(G) contains an element of order p which preserves no edges. In this short note, we investigate the behavior of graph polynomials (Negami and Tutte) with respect to graph periodicity. In particular, we prove that if p is a prime, then the coefficients of the Tutte polynomial of such a graph satisfy a certain necessary condition.

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Forgotten Index of Generalized Operations on Graphs

Journal of Chemistry ◽

10.1155/2021/9971277 ◽

2021 ◽

Vol 2021 ◽

pp. 1-14

Author(s):

Muhammad Javaid ◽

Saira Javed ◽

Saima Q. Memon ◽

Abdulaziz Mohammed Alanazi

Keyword(s):

Structural Properties ◽

Topological Indices ◽

Theoretical Chemistry ◽

Factor Graphs ◽

Molecular Graphs ◽

Strong Product ◽

Forgotten Index ◽

Counting Polynomial ◽

Product Of Graphs

In theoretical chemistry, several distance-based, degree-based, and counting polynomial-related topological indices (TIs) are used to investigate the different chemical and structural properties of the molecular graphs. Furtula and Gutman redefined the F -index as the sum of cubes of degrees of the vertices of the molecular graphs to study the different properties of their structure-dependency. In this paper, we compute F -index of generalized sum graphs in terms of various TIs of their factor graphs, where generalized sum graphs are obtained by using four generalized subdivision-related operations and the strong product of graphs. We have analyzed our results through the numerical tables and the graphical presentations for the particular generalized sum graphs constructed with the help of path (alkane) graphs.

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Geometric structural properties of bonded layers of chemically modified silicas

Journal of Chromatography A ◽

10.1016/s0021-9673(01)91460-8 ◽

1988 ◽

Vol 447 (3) ◽

pp. 103-116 ◽

Cited By ~ 2

Author(s):

A Fadeev

Keyword(s):

Structural Properties ◽

Chemically Modified ◽

Chemically Modified Silicas

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Electronic, vibrational and structural properties of alkali halide clusters

Physica B Condensed Matter ◽

10.1016/s0921-4526(84)92608-5 ◽

1984 ◽

Vol 127 (1-3) ◽

pp. 214-218 ◽

Cited By ~ 14

Author(s):

T MARTIN

Keyword(s):

Structural Properties ◽

Alkali Halide

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Newborn's Preference for Faces

European Psychologist ◽

10.1027/1016-9040.1.3.200 ◽

1996 ◽

Vol 1 (3) ◽

pp. 200-205 ◽

Cited By ~ 12

Author(s):

Carlo Umiltà ◽

Francesca Simion ◽

Eloisa Valenza

Keyword(s):

Visual System ◽

Structural Properties ◽

Sensory Properties ◽

Human Face ◽

System Experiment ◽

Face Experiment

Four experiments were aimed at elucidating some aspects of the preference for facelike patterns in newborns. Experiment 1 showed a preference for a stimulus whose components were located in the correct arrangement for a human face. Experiment 2 showed a preference for stimuli that had optimal sensory properties for the newborn visual system. Experiment 3 showed that babies directed their attention to a facelike pattern even when it was presented simultaneously with a non-facelike stimulus with optimal sensory properties. Experiment 4 showed the preference for facelike patterns in the temporal hemifield but not in the nasal hemifield. It was concluded that newborns' preference for facelike patterns reflects the activity of a subcortical system which is sensitive to the structural properties of the stimulus.

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