Omega polynomial, counting opposite edge strips ops, was proposed by Diudea to describe cycle-containing molecular structures, particularly those associated with nanostructures. In this paper, some theoretical aspects are evidenced and particular cases are illustrated.

We generalize the notion of quandle polynomial to case singquandles. show that singquandle is an invariant finite also construct a singular link from and this new generalizes counting invariant. In particular, using invariant, we can distinguish links with same

Tiling planar regions with dominoes is a classical problem in which the decision and counting problems are polynomial. We prove a variety of hardness results (both NPand #Pcompleteness) for different generalizations of dominoes in three and higher dimensions.

We present a transformation from longest paths to shortest paths in sub-classes of directed acyclic graphs (DAGs). The transformation needs log-space and oracle access to reachability in the same class of graphs. As a corollary, we obtain our main result: Longest Paths in planar DAGs is in UL ∩ co-UL. The result also extends to toroidal DAGs. Further, we show that Longest Paths in max-unique DA...

It is known due to the work of Van den Broeck, Meert and Darwiche that weighted first-order model counting (WFOMC) in two-variable fragment logic can be solved time polynomial number domain elements. In this paper we extend result with quantifiers.

We prove #P-completeness for counting the number of forests in regular graphs and chordal graphs. We also present algorithms for this problem, running in O∗(1.8494m) time for 3-regular graphs, and O∗(1.9706m) time for unit interval graphs, where m is the number of edges in the graph and O∗-notation ignores a polynomial factor. The algorithms can be generalized to the Tutte polynomial computation.

We prove #P-completeness for counting the number of forests in regular graphs and chordal graphs. We also present algorithms for this problem, running in O∗(1.8494m) time for 3-regular graphs, and O∗(1.9706m) time for unit interval graphs, where m is the number of edges in the graph and O∗-notation ignores a polynomial factor. The algorithms can be generalized to the Tutte polynomial computation.

We introduce several new methods to obtain upper bounds on the number of solutions of the congruences f(x) ≡ y (mod p) and f(x) ≡ y (mod p), with a prime p and a polynomial f , where (x, y) belongs to an arbitrary square with side length M . We give two applications of these results to counting hyperelliptic curves in isomorphism classes modulo p and to the diameter of partial trajectories of a...

We consider approximate counting of colorings of an n-vertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colorings would mix rapidly, provided the number of colors k exceeded the maximum degree ∆ of the graph by a factor of at least 2. We prove that this is not a necessary condition for rapid mixing by consideri...

Cook and Reckhow proved in 1979 that the propositional pigeonhole principle has polynomial size extended Frege proofs. Buss proved in 1987 that it also has polynomial size Frege proofs; these Frege proofs used a completely different proof method based on counting. This paper shows that the original Cook and Reckhow extended Frege proofs can be formulated as quasipolynomial size Frege proofs. Th...