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Target class classification is a mixed classification and transition model whose integrated goal is to assign objects to a certain, so called target or normal class. The classification process is iterative, and in each step an object in a…
Given a subspace arrangement, there are several De Concini-Procesi models associated to it, depending on distinct sets of initial combinatorial data (building sets). The first goal of this paper is to describe, for the root arrangements of…
The motivation comes from boson fermion correspondence. This article shows for each fermionic root there is correspondence a bosonic root, as a result we get for each Dynkin diagram of Lie Superalgebra a corresponding flip Dynkin…
We consider the symmetric difference of two graphs on the same vertex set $[n]$, which is the graph on $[n]$ whose edge set consists of all edges that belong to exactly one of the two graphs. Let $\mathcal{F}$ be a class of graphs, and let…
We give a criterion for a Dynkin diagram, equivalently a generalized Cartan matrix, to be symmetrizable. This criterion is easily checked on the Dynkin diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of compact…
An analysis is given of the structure of a general two-dimensional Toda field theory involving bosons and fermions which is defined in terms of a set of simple roots for a Lie superalgebra. It is shown that a simple root system for a…
We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group $S_{2n+1}$. We define two classes of moves on such permutations, called trivial petal additions and crossing…
The algebraic approach to the construction of the reflexive polyhedra that yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres reveals graphs that include and generalize the Dynkin diagrams associated with gauge…
Let $G$ be a signed graph. Let $\hat{G}$ be the graph obtained from $G$ by replacing each edge $e$ by a chain or a sheaf. We first establish a relation between the $Q$-polynomial of $\hat{G}$[6] and the $W$-polynomial of $G$ [9]. Two…
We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the $d$-dimensional polytopes contained in $\mathbb{R}^d$ and its edges connect any two polytopes that can be obtained from one another by either…
In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For…
This talk was presented at Workshop "Spectral Methods in Representation Theory of Algebras and Applications to the Study of Rings of Singularities", 2008 (Banff, Canada). W. Ebeling established a connection between certain Poincare series,…
Let $\Gamma$ be the graph on the roots of the $E_8$ root system, where any two distinct vertices $e$ and $f$ are connected by an edge with color equal to the inner product of $e$ and $f$. For any set $c$ of colors, let $\Gamma_c$ be the…
Networks embedded in space can display all sorts of transitions when their structure is modified. The nature of these transitions (and in some cases crossovers) can differ from the usual appearance of a giant component as observed for the…
Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural "Coxeter…
In this article we consider commuting graphs of involution conjugacy classes in the affine Weyl group of type $\tilde C_n$. We show that where the graph is connected the diameter is at most $n+2$.
Scaffolds are certain tensors arising in the study of association schemes, and have been (implicitly) understood diagrammatically as digraphs with distinguished "root" nodes and with matrix edge weights, often taken from Bose-Mesner…
We introduce the notion of a move graph, that is, a directed graph whose vertex set is a $\mathbb Z$-module $\mathbb Z_n^m$, and whose arc set is uniquely determined by the action $M\!:\!\mathbb Z_n^m\to \mathbb Z_n^m$ where $M$ is an…
Let $W(D_n)$ denote the walk matrix of the Dynkin graph $D_n$, a tree obtained from the path of order $n-1$ by adding a pendant edge at the second vertex. We prove that $\text{rank}\,W(D_n)=n-2$ if $4\mid n$ and $\text{rank}\,W(D_n)=n-1$…
In the present note, we complete the correspondence between stratum components of translation surfaces in low genus and finite-type Artin groups with defining Dynkin diagram containing $E_6$. In an earlier work, we showed that in genus $3$…