Chemical applications of topology and group theory

The 60 even permutations of the ligands in the five-coordinate complexes, ML ₅, form the alternating group A ₅, which is isomorphic with the icosahedral pure rotation group I. Using this idea, it is shown how a regular icosahedron can be used as a topological representation for isomerizations of the five-coordinate complexes, ML ₅, involving only even permutations if the five ligands L correspond either to the five nested octahedra with vertices located at the midpoints of the 30 edges of the icosahedron or to the five regular tetrahedra with vertices located at the midpoints of the 20 faces of the icosahedron. However, the 120 total permutations of the ligands in five-coordinate complexes ML ₅ cannot be analogously represented by operations in the full icosahedral point group I _h, since I _his the direct product I×C₂ whereas the symmetric group S ₅ is only the semi-direct product A ₅S₂. In connection with previously used topological representations on isomerism in five-coordinate complexes, it is noted that the automorphism groups of the Petersen graph and the Desargues-Levi graph are isomorphic to the symmetric group S ₅ and to the direct product S ₅×S ₂, respectively. Applications to various fields of chemistry are briefly outlined.     

Improved square coloring of planar graphs

Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac{3}{2}\mathrm{\Delta }+1$ colors are sufficient to square color every planar graph of maximum degree Δ. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that $2\mathrm{\Delta }+7$ colors are sufficient, which improves the best known bounds when $6?\mathrm{\Delta }?31$.     

Locally s-distance transitive graphs and pairwise transitive designs

The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs with a star quotient is equivalent to the study of a particular family of designs with strong symmetry properties that we call nicely affine and pairwise transitive. We show that a group acting regularly on the points of such a design must be abelian and give general construction for this case.     

Binding numbers for fractional ID-k-factor-critical graphs

LetG be a graph,and k≥2 be a positive integer.A graph G is fractional independentset-deletable k-factor-critical(in short,fractional ID-k-factor-critical),if G I has a fractional k-factor for every independent set I of G.The binding number bind(G)of a graph G is defined as bind(G)=min|NG(X)||X|:=X V(G),NG(X)=V(G).In this paper,it is proved that a graph G is fractional ID-k-factor-critical if n≥6k 9 and bind(G)(3k 1)(n 1)kn 2k+2.     

Continuum versus discrete networks,graph Laplacians,and reproducing kernel Hilbert spaces

Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space $\mathfrak{X}$. We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of $L^2$ functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example).     

Characterizing and Generating Bivariate Empirical Rank Distributions Satisfying Certain Positive Dependence Concepts

Abstract

This article introduces an approach for characterizing the classes of empirical distributions that satisfy certain positive dependence notions. Mathematically, this can be expressed as studying certain subsets of the class S_N of permutations of 1, …, N, where each subset corresponds to some positive dependence notions. Explicit techniques for it-eratively characterizing subsets of S_N that satisfy certain positive dependence concepts are obtained and various counting formulas are given. Based on these techniques, graph-theoretic methods are used to introduce new and more efficient algorithms for constructively generating and enumerating the elements of various of these subsets of S_N. For example, the class of positively quadrant dependent permutations in S_N is characterized in this fashion.     

Novel matrix invariants for characterization of changes of proteomics maps

Previous studies on mathematical characterization of proteomics maps by sets of map invariants were based on the construction of a set of distance-related matrices obtained by matrix multiplication of a single matrix by itself. Here we consider an alternative characterization of proteomics maps based on a set of matrices characterizing local features of an embedded zigzag curve over the map. It is shown that novel invariants can well characterize proteomics maps. Advantages of the novel approach are discussed.