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.     

Leaf‐Critical and Leaf‐Stable Graphs

The minimum leaf number ml(G) of a connected graph G is defined as the minimum number of leaves of the spanning trees of G if G is not hamiltonian and 1 if G is hamiltonian. We study nonhamiltonian graphs with the property for each or for each . These graphs will be called ‐leaf‐critical and l‐leaf‐stable, respectively. It is far from obvious whether such graphs exist; for example, the existence of 3‐leaf‐critical graphs (that turn out to be the so‐called hypotraceable graphs) was an open problem until 1975. We show that l‐leaf‐stable and l‐leaf‐critical graphs exist for every integer , moreover for n sufficiently large, planar l‐leaf‐stable and l‐leaf‐critical graphs exist on n vertices. We also characterize 2‐fragments of leaf‐critical graphs generalizing a lemma of Thomassen. As an application of some of the leaf‐critical graphs constructed, we settle an open problem of Gargano et al. concerning spanning spiders. We also explore connections with a family of graphs introduced by Grünbaum in correspondence with the problem of finding graphs without concurrent longest paths.     

Partite Saturation Problems

We look at several saturation problems in complete balanced blow‐ups of graphs. We let denote the blow‐up of H onto parts of size n and refer to a copy of H in as partite if it has one vertex in each part of . We then ask how few edges a subgraph G of can have such that G has no partite copy of H but such that the addition of any new edge from creates a partite H. When H is a triangle this value was determined by Ferrara, Jacobson, Pfender, and Wenger in  5 . Our main result is to calculate this value for when n is large. We also give exact results for paths and stars and show that for 2‐connected graphs the answer is linear in n whilst for graphs that are not 2‐connected the answer is quadratic in n. We also investigate a similar problem where G is permitted to contain partite copies of H but we require that the addition of any new edge from creates an extra partite copy of H. This problem turns out to be much simpler and we attain exact answers for all cliques and trees.     

Maximizing H‐Colorings of Connected Graphs with Fixed Minimum Degree

For graphs G and H , an H‐coloring of G is a map from the vertices of G to the vertices of H that preserves edge adjacency. We consider the following extremal enumerative question: for a given H , which connected n‐vertex graph with minimum degree δ maximizes the number of H‐colorings? We show that for nonregular H and sufficiently large n , the complete bipartite graph is the unique maximizer. As a corollary, for nonregular H and sufficiently large n the graph is the unique k‐connected graph that maximizes the number of H‐colorings among all k‐connected graphs. Finally, we show that this conclusion does not hold for all regular H by exhibiting a connected n‐vertex graph with minimum degree δ that has more ‐colorings (for sufficiently large q and n ) than .