New Polynomial-Based Molecular Descriptors with Low Degeneracy

In this paper, we introduce a novel graph polynomial called the ‘information polynomial’ of a graph. This graph polynomial can be derived by using a probability distribution of the vertex set. By using the zeros of the obtained polynomial, we additionally define some novel spectral descriptors. Compared with those based on computing the ordinary characteristic polynomial of a graph, we perform a numerical study using real chemical databases. We obtain that the novel descriptors do have a high discrimination power

Transit sets of two-point crossover

Genetic Algorithms typically invoke crossover operators to produce offsprings that are a “mixture” of two parents x and y. On strings, k-point crossover breaks parental genotypes at at most k corresponding positions and concatenates alternating fragments for the two parents. The transit set Rk(x, y) comprises all offsprings of this form. It forms the tope set of an uniform oriented matroid with Vapnik-Chervonenkis dimension k + 1. The Topological Representation Theorem for oriented matroids thus implies a representation in terms of pseudosphere arrangements. This makes it possible to study 2-point crossover in detail and to characterize the partial cubes defined by the transit sets of two-point crossover.Department of Science and Technology of India (SERB project file no. MTR/2017/000238 “Axiomatics of betweenness in discrete structures” to MC), and the German Academic Exchange Service (DAAD) through the bilateral Slovenian-German project “Mathematical Foundations of Selected Topics in Science”