A Graph Polynomial Approach to Primitivity

2013 ◽  
pp. 153-164 ◽  
Author(s):  
Francine Blanchet-Sadri ◽  
Michelle Bodnar ◽  
Nathan Fox ◽  
Joe Hidakatsu
Keyword(s):  
Graph Polynomial

scholarly journals A Graph Polynomial for Independent Sets of Bipartite Graphs

2012 ◽  
Vol 21 (5) ◽  
pp. 695-714 ◽  
Author(s):  
Q. GE ◽  
D. ŠTEFANKOVIČ
Keyword(s):  
Riemann Hypothesis ◽  
Independent Sets ◽  
Bipartite Graphs ◽  
Rational Points ◽  
Perfect Matchings ◽  
Generalized Riemann Hypothesis ◽  
Graph Polynomial ◽  
Exact Evaluation

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS).We analyse the complexity of exact evaluation of the polynomial at rational points and show a dichotomy result: for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial.

scholarly journals KHOVANOV–ROZANSKY GRAPH HOMOLOGY AND COMPOSITION PRODUCT

2008 ◽  
Vol 17 (12) ◽  
pp. 1549-1559 ◽  
Author(s):  
E. WAGNER
Keyword(s):  
Recursive Formula ◽  
Graph Polynomial ◽  
Graph Homology ◽  
Composition Product

In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov–Rozansky graph homology.

scholarly journals General Form of Domination Polynomial for Two Types of Graphs Associated to Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 149-155
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Complete Graphs ◽  
Dominating Sets ◽  
Dihedral Groups ◽  
Graph Polynomial ◽  
Central Elements ◽  
Domination Polynomial ◽  
Class Graph

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups.

scholarly journals A weighted graph polynomial from chromatic invariants of knots

1999 ◽  
Vol 49 (3) ◽  
pp. 1057-1087 ◽  
Author(s):  
Steven D. Noble ◽  
Dominic J. A. Welsh
Keyword(s):  
Weighted Graph ◽  
Graph Polynomial

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

2012 ◽  
Vol 21 (10) ◽  
pp. 1250098 ◽  
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.

scholarly journals A graph polynomial and its applications

1972 ◽  
Vol 3 (4) ◽  
pp. 305-313 ◽  
Author(s):  
Frank H. Clarke
Keyword(s):  
Graph Polynomial
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