The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobás–Riordan– Tutte polynomial generalizes the Tutte polynomial of graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any lin...

For each graph and each positive integer n, we define a chain complex whose graded Euler characteristic is equal to an appropriate nspecialization of the dichromatic polynomial. This also gives a categorification of n-specializations of the Tutte polynomial of graphs. Also, for each graph and integer n ≤ 2, we define the different one variable n-specializations of the dichromatic polynomials, a...

The terminal matrix is the distance matrix between all pairs of valence one vertices of a given graph and the terminal polynomial is the characteristic polynomial of the terminal matrix. Star-like graphs are connected graphs with at most one vertex of degree 3 d  . The recursive formula that calculates the terminal polynomial of a star-like graph is given. Particularly, the formula that calcul...

Negami and Kawagoe has already defined a polynomial f(G) associated with each graph G as what discriminates graphs more finely than the polynomial f(G) definedby Negami and the Tutte polynomial. In this paper, we shall show that the polynomial f(G) includes potentially the generating function counting the independent sets and the degree sequence of a graph G, which cannot be recognized from f(G...

We investigate combinatorial properties of a graph polynomial indexed by halfedges of a graph which was introduced recently to understand the connection between Feynman rules for scalar field theory and Feynman rules for gauge theory. We investigate the new graph polynomial as a stand-alone object.

Let G be a connected graph. We say that given graph is tree if every pair ofvertices by unique path. The rooted product of two trees relevant to tree, as the obtained another tree. In this paper, we establish independent neighborhood sets and obtain its corresponding polynomial. Furthermore, polynomial were determine using their sets.

The Tutte polynomial of a graph, also known as the partition function of the q-state Potts model, is a 2-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chromatic polynomial and flow polynomial as partial evaluations, and various numerical invariants, such as the number of ...

We present a novel approach for preprocessing systems of polynomial equations via graph partitioning. The variable-sharing graph of a system of polynomial equations is defined. If such graph is disconnected, then the corresponding system of equations can be split into smaller ones that can be solved individually. This can provide a tremendous speed-up in computing the solution to the system, bu...

We introduce the concept of a relative Tutte polynomial. We show that the relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (hence the Jones polynomial) of a virt...

Modifying a given graph to obtain another graph is a wellstudied problem with applications in many fields. Given two input graphs G and H, the Contractibility problem is to decide whether H can be obtained from G by a sequence of edge contractions. This problem is known to be NP-complete already when both input graphs are trees of bounded diameter. We prove that Contractibility can be solved in...