design of crystal-like lattices can be achieved by using some net operations. hypothetical networks, thus obtained, can be characterized in their topology by various counting polynomials and topological indices derived from them. the networks herein presented are related to the dyck graph and described in terms of omega polynomial and piv polynomials.

The Markov-Dyck shifts arise from finite directed graphs. An expression for the zeta function of a Markov-Dyck shift is given. The derivation of this expression is based on a formula in Keller (G. Keller, Circular codes, loop counting, and zeta-functions, J. Combinatorial Theory 56 (1991), pp. 75– 83). For a class of examples that includes the Fibonacci-Dyck shift the zeta functions and topolog...

Dynamic complexity is concerned with updating the output of a problem when the input is slightly changed. We study the dynamic complexity of Dyck reachability problems in directed and undirected graphs, where updates may add or delete edges. We show a strong dichotomy between such problems, based on the size of the Dyck alphabet. Some of them are P-complete (under a strong notion of reduction) ...

We consider the parameter rank introduced for graph configurations by M. Baker and S. Norine. We focus on complete graphs and obtain an efficient algorithm to determine the rank for these graphs. The analysis of this algorithm leads to the definition of a parameter on Dyck words, which we call prerank. We prove that the distribution of area and prerank on Dyck words of given length 2n leads to ...

Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue chromatic quasisymmetric function called chromatic polynomial Dyck graph. We give positive expansion for this in basis fundamental slide polynomials using recent work Assaf-Bergeron on flagged <mml:math xmlns:mml="ht...

A {0, 1}-matrix A is balanced if it does not contain a submatrix of odd order having exactly two 1’s per row and per column. A graph is balanced if its clique-matrix is balanced. No characterization of minimally unbalanced graphs is known, and even no conjecture on the structure of such graphs has been posed, contrarily to what happened for perfect graphs. In this paper, we provide such a chara...

A Dyck path with 2k steps and e flaws is a path in the integer lattice that starts at the origin and consists of k many ↗-steps and k many ↘-steps that change the current coordinate by (1, 1) or (1,−1), respectively, and that has exactly e many ↘-steps below the line y = 0. Denoting by D 2k the set of Dyck paths with 2k steps and e flaws, the Chung-Feller theorem asserts that the sets D 2k, D 1...

Abstract We present a new sufficient criterion to prove that non-sofic half-synchronized subshift is direct prime. The based on conjugacy invariant properties of Fischer graphs shifts. use this show as result all n -Dyck shifts are prime, and we also give proofs primeness beta-shifts S -gap construct class synchronized prime subshifts which additionally admit reversible cellular automata with d...