Many polynomial invariants are defined on graphs for encoding the combinatorial information and researching them algebraically. In this paper, we introduce the cycle polynomial and the path polynomial of directed graphs for counting cycles and paths, respectively. They satisfy recurrence relations with respect to elementary edge or vertex operations. They are related to other polynomials and ca...

The notions separability and normality are related to this characterisation. In the case of skew fields polynomials often have infinitely many zeros, so a different way of counting zeros as distinct is needed. The well-known theorem of Gordon and Motzkin [Z] states that a polynomial of degree n has zeros in at most n conjugacy classes. This suggests one should count zeros of a polynomial by the...

Consider an undirected Eulerian graph, a graph in which each vertex has even degree. An Eulerian orientation of the graph is an orientation of its edges such that for each vertex v, the number of incoming edges of v equals to outgoing edges of v, i.e. din(v) = dout(v). Let P0 denote the set of all Eulerian orientations of graph G. In this paper, we are concerned with the questions of sampling u...

A {+,×}-circuit counts a given multivariate polynomial f , if its values on 0-1 inputs are the same as those of f ; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 0-1 input vector (with multiplicities given by their coefficients). A circuit decides f if it has the same 0-1 roots as f . We first show that some...

Figure 1: A length-18 in the complete 2D grid. lems in different areas of science, such as combinatorics, statistical physics, theoretical chemistry, and computer science. By “twodimensional grid” we mean the two-dimensional rectangular lattice Z2 with origin (0, 0). One of the most prominent applications of counting SAWs is the modeling of spatial arrangement of linear polymer molecules in a s...

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...

Recently it has been shown that elements of the unitary matrix determined by a quantum circuit can be computed by counting the number of common roots in the ˇnite ˇeld Z2 for a certain set of multivariate polynomials over Z2. In this paper we present a C# package that allows a user to assemble a quantum circuit and to generate the multivariate polynomial set associated with the circuit. The gen...