Related papers: The Dirac operator of a graph

We study a self-adjoint realization of a massless Dirac operator on a bounded connected domain $\Omega\subset \mathbb{R}^2$ which is frequently used to model graphene quantum dots. In particular, we show that this operator is the limit, as…

For a group $G$, the generating graph $\Gamma(G)$ is defined as the graph with the vertex set $G$, and any two distinct vertices of $\Gamma(G)$ are adjacent if they generate $G$. In this paper, we study the generating graph of $D_n,$ where…

For a representation of a Lie algebra, one can construct a diagram of the representation, i. e. a directed graph with edges labeled by matrix elements of the representation. This article explains how to use these diagrams to describe normal…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $L(G)$ be the line graph of $G$, which has vertex set $E(G)$ and two vertices $e$ and $f$ of $L(G)$ is adjacent if $e$ and $f$ is incident in $G$. The vertex-edge graph…

In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A diclique of a digraph is a pair $V \to W$ of sets of vertices such that $v…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$…

We use a continued fraction expansion of the sign-function in order to obtain a five dimensional formulation of the overlap lattice Dirac operator. Within this formulation the inverse of the overlap operator can be calculated by a single…

We prove that the C*-algebra of a directed graph $E$ is liminal iff the graph satisfies the finiteness condition: if $p$ is an infinite path or a path ending with a sink or an infinite emitter, and if $v$ is any vertex, then there are only…

In this largely expository paper we give a self-contained treatment of the Dirac operator. Emphasizing the algebraic point of view we first sketch the necessary prerequisites from Clifford algebras and their representations and then define…

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by a trivial automorphism. A list assignment to $G$ is an assignment $L = \{L(v)\}_{v\in V…

We study the set of all determinants of adjacency matrices of graphs with a given number of vertices.

This paper is a follow-up on the \emph{noncommutative differential geometry on infinitesimal spaces} [15]. In the present work, we extend the algebraic convergence from [15] to the geometric setting. On the one hand, we reformulate the…

Given any digraph $D$ on $n$ vertices, let $\mathcal{P}(D)$ be the family of all directed paths in $D$, and let $H$ be a digraph with the arc set $A(H)=\{a_1, \ldots, a_k\}$. The digraph $D$ is called arbitrary Hamiltonian $H$-linked if for…

Given a commuting d-tuple $\bar T=(T_1,...,T_d)$ of otherwise arbitrary nonnormal operators on a Hilbert space, there is an associated Dirac operator $D_{\bar T}$. Significant attributes of the d-tuple are best expressed in terms of…

We are using the finite-gap approach for the construction of the Schr\"{o}dinger operator discretization on a quad graph. The latter is represented by a two-dimensional integer sublattice in a $d$-dimensional space. The Green's function of…

We study equivariant families of Dirac operators on the source fibers of a Lie groupoid with a closed space of units and equipped with an action of an auxiliary compact Lie group. We use the Getzler rescaling method to derive a fixed-point…

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of A, D, E-type. In this paper we…

Let $G=(V,E)$ be a graph. A set $S\subseteq V(G)$ is a dominating set, if every vertex in $V(G)\backslash S$ is adjacent to at least one vertex in $S$. The $k$-dominating graph of $G$, $D_k (G)$, is defined to be the graph whose vertices…

A digraph D is the pattern of a matrix M when D has an arc ij if and only if the ij-th entry of M is nonzero. Study the relationship between unitary matrices and their patterns is motivated by works in quantum chaology and quantum…

Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $\delta(G) \geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex…