Topological chiral modes in random scattering networks

Using elementary graph theory, we show the existence of interface chiral modes in random oriented scattering networks and discuss their topological nature. For particular regular networks (e.g. L-lattice, Kagome and triangular networks), an explicit mapping with time-periodically driven (Floquet) tight-binding models is found. In that case, the interface chiral modes are identified as the celebrated anomalous edge states of Floquet topological insulators and their existence is enforced by a symmetry imposed by the associated network. This work thus generalizes these anomalous chiral states beyond Floquet systems, to a class of discrete-time dynamical systems where a periodic driving in time is not required.

Heuristic method to determine lucky k-polynomials for k-colorable graphs

The existence of edges is a huge challenge with regards to determining lucky k-polynomials of simple connected graphs in general. In this paper the lucky 3-polynomials of path and cycle graphs of order, 3 ≤ n ≤ 8 are presented as the basis for the heuristic method to determine the lucky k-polynomials for k-colorable graphs. The difficulty of adjacency with graphs is illustrated through these elementary graph structures. The results are also illustratively compared with the results for null graphs (edgeless graphs). The paper could serve as a basis for finding recurrence results through innovative methodology.

First steps to literacy: from drawing to writing

Materials of the manual are relevant and timely methodical literature. The relevance is due to the importance of preparing children with disabilities to master the basic educational activities. This manual can serve as one of the preparatory stages of the formation of children with ovz elementary graph-motor skills for successful mastery of the initial skills of writing. The advantage of this work is its appeal to the needs of practice, detailed elaboration of the specifics of the development of children's graphic skills, as well as the availability of the proposed tasks for children with special educational needs.

Comparison of hydrogen bonds and diverse weak interactions of the nitro group in 2-methyl-4-nitroanilinium nitrate, bisulfate and two hexafluoridosilicates: elementary graph-set approach

Crystal structures of (H2m4na)NO3(1), (H2m4na)HSO4(2), (H2m4na)2SiF6(3) and (H2m4na)2SiF6·2H2O (4), where 2m4na = 2-methyl-4-nitroaniline, are presented. Two layers of interactions occur in the structures, N—H...O/F hydrogen bonds and interactions with the nitro group. Although diverse, hydrogen-bonding patterns are compared with each other by means of interrelations among elementary graph-set descriptors and descriptors of hydrogen-bonding patterns. Using mathematical relations, the gradual expansion of the ring patterns was shown in the crystal structures. Parallel and perpendicular arranged nitro groups form weak π(N)NO2...π(O)NO2and π(N)NO2...ONO2interactions, respectively. The πNO2...πringinteraction has an impact to the stabilization of parallel oriented nitro groups. Generally, weak interactions constructed by the nitro group occur in the studied crystals as follows: π(N)NO2...π(O)NO2, πring...πring, C—H...O (1); π(N)NO2...π(O)NO2, π(N)NO2...ONO2(2); π(N)NO2...π(O)NO2, π(N)NO2...ONO2(3); C—H...O (4).

On the Optimality of Conservation Results for Local Reflection in Arithmetic

Let T be a recursively enumerable theory extending Elementary Arithmetic EA. L. D. Beklemishev proved that the Σ2 local reflection principle for T, (T), is conservative over the Σ1 local reflection principle, (T), with respect to boolean combinations of Σ1-sentences; and asked whether this result is best possible. In this work we answer Beklemishev's question by showing that Π2-sentences are not conserved for T = EA + “f is total,” where f is any nondecreasing computable function with elementary graph. We also discuss how this result generalizes to n > 0 and obtain as an application that for n > 0, is conservative over IΣn with respect to Πn+2-sentences.

Visual reasoning with graph-based mechanisms: the good, the better and the best

This paper presents a graph-based knowledge representation and reasoning language. This language benefits from an important syntactic operation, which is called a graph homomorphism. This operation is sound and complete with respect to logical deduction. Hence, it is possible to do logical reasoning without using the language of logic but only graphical, thus visual, notions. This paper presents the main knowledge constructs of this language, elementary graph-based reasoning mechanisms, as well as the graph homomorphism, which encompasses all these elementary transformations in one global step. We put our work in context by presenting a concrete semantic annotation application example.

PERFECT MATCHINGS AND GENUS OF SOME CARTESIAN PRODUCTS OF GRAPHS

Lovász and Plummer conjectured that: for k ≥ 3 there exist constants c1(k) > 1 and c2(k) > 0 such that every k-regular elementary graph on 2n vertices, without forbidden edges, contains at least c2(k)⋅c1(k)nperfect matchings. Furthermore c1(k) → ∞ as k → ∞. In this paper, for some Cartesian products of graphs, we obtain a lower bound of their number of perfect matchings which is similar to that of Lovász and Plummer's conjecture. Furthermore, we compute the genus of some Cartesian products of graphs.