Parabolic subgroups in FC-type Artin groups

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results of Cumplido, Gebhardt, Gonzales-Meneses and Wiest, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC-type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed by Cumplido et al. using parabolic subgroups. We extend the construction of this complex, called the complex of parabolic subgroups, to FC-type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is δ-hyperbolic.     

Hermitian adjacency matrix of the second kind for mixed graphs

This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind (N-matrix for short) introduced by Mohar [25]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the sixth root of unity $\omega =\frac{1+\mathbf{i}\sqrt{3}}{2}$ (and its symmetric entry is $\overline{\omega }=\frac{1?\mathbf{i}\sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: equivalent conditions for a mixed graph that shares the same spectrum of its N-matrix with its underlying graph are given. A sharp upper bound on the spectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called two-way and three-way switchings are discussed–they give rise to some cospectral mixed graphs. We extract all the mixed graphs whose rank of its N-matrix is 2 (resp. 3). Furthermore, we show that if $M_G$ is a connected mixed graph with rank 2, then $M_G$ is switching equivalent to each connected mixed graph to which it is cospectral. However, this does not hold for some connected mixed graphs with rank 3. We identify all mixed graphs whose eigenvalues of its N-matrix lie in the range $(?\alpha ,\alpha )$ for $\alpha \in \{\sqrt{2},\sqrt{3},2\}$.     

On distance-regular Cayley graphs of generalized dicyclic groups

Let G be a generalized dicyclic group with identity 1. An inverse closed subset S of $G?\{1\}$ is called minimal if $〈S〉=G$ and there exists some $s\in S$ such that $〈S?\{s,s^{?1}\}〉\ne G$. In this paper, we characterize distance-regular Cayley graphs $\mathrm{Cay}(G,S)$ of G under the condition that S is minimal.     

Sequence-dependent clustering properties of nucleotides fragments in an ionic solution

The chemical nature of the DNA bases is an important factor in sequence-mediated association of DNA molecules. Nucleotides are the fundamental DNA elements and the base identity impacts the molecular properties of nucleotide fragments. It is interesting to study the fundamental nature of nucleotides in DNA, on the basis of base-specific interactions, association, and modes of standard atomic or molecular interactions. With all-atom molecular dynamics simulations of model dinucleotide and tetranucleotide systems having single-stranded dinucleotide or tetranucleotide fragments of varying sequences, we show how the base identity and interactions between the different bases as well as water may affect the clustering properties of nucleotides fragments in an ionic solution. Sequence-dependent differential interactions between the nucleotide fragments, ionic concentration, and elevated temperature are found to influence the clustering properties and dynamics of association. Well-known epigenetic modification of DNA, that is, cytosine methylation also promotes dinucleotide clustering in solution. These observations point to one possible chemical nature of the DNA bases, as well as the importance of the base pairing, base stacking, and ionic interactions in DNA structure formation, and DNA sequence-mediated association. Sequence- and the ionic environment-mediated self-association properties of the dinucleotides indicate its great potential to develop biological nanomaterials for desired applications.