In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators Tφ with trigonometric polynomial symbols φ. Our criterion involves the zeros of an analytic polynomial f induced by the Fourier coefficients of φ. Moreover the rank of the selfcommutator of Tφ is computed from the number of zeros of f in the...

A topological index of a graph G is a numeric quantity related to G which is invariant under automorphisms of G. A new counting polynomial, called the "Omega" W(G, x) polynomial, was recently proposed by Diudea on the ground of quasi-orthogonal cut "qoc" edge strips in a polycyclic graph. In this paper, the vertex PI, Szeged and omega polynomials of carbon nanocones CNC4[n] are computed.

We prove an eeective mean-value theorem for the values of a non-degenerate, algebraic exponential polynomial in several variables. These objects simultaneously generalise the fundamental examples of linear recurrence sequences and sums of S-units. The proof is based on an eeective, uniform estimate for the deviation of the exponential polynomial from its expected value. This estimate is also us...

Hamming graphs are Cartesian products of complete graphs and partial Hamming graphs are their isometric subgraphs. The Hamming polynomial h(G) of a graph G is introduced as the Hamming subgraphs counting polynomial. Kk-derivates ∂kG (k ≥ 2) of a partial Hamming graph are also introduced. It is proved that for a partial Hamming graph G, ∂h(G) ∂xk = h(∂kG). A couple of combinatorial identities in...

Consider the field F2n . Its elements are presented as polynomials from F2[x] modulo some irreducible polynomial of degree n. This polynomial can be found in time polynomial in n, as well as the matrix that related two representation corresponding to different irreducible polynomials [2]. Therefore, we do not need to specify a choice of the irreducible polynomial speaking about polynomial reduc...

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel by proving a GapL ⊆ NC ⊆ P upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded clique-width graphs can be performed efficiently in parallel. Thus we show...

Let G = (U; V; E) be a bipartite graph with jUj = jV j = n. The factor size of G, f, is the maximum number of edge disjoint perfect matchings in G. We characterize the complexity of counting the number of perfect match-ings in classes of graphs parameterized by factor size. We describe the simple algorithm, which is an approximation algorithm for the permanent that is a natural simpliication of...

Counting the independent sets of a graph is classical #P-complete problem, even in bipartite case. We give an exponential-time approximation scheme for this problem which faster than best known algorithm exact problem. The running time our on general graphs with error tolerance ε at most O(20.2680n) times polynomial 1/ε. On graphs, exponential term improved to O(20.2372n). Our methods combine t...

In this thesis we develop FPTASs for the counting problems of m−tuples, contingency tables with two rows, and 0/1 knapsack. For the problem of counting m−tuples, we design two algorithms, one is strongly polynomial. As far as we know, these are the first FPTASs for this problem. For the problem of counting contingency tables we improve significantly over the running time of existing algorithms....

The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k ≥ 0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonac...