Related papers: Enumerating the Classes of Local Equivalency in Gr…
An identifying code in a graph is a set of vertices which intersects all the symmetric differences between pairs of neighbourhoods of vertices. Not all graphs have identifying codes; those that do are referred to as twin-free. In this…
We bring together in one place some of the main results and applications from our recent works in quantum information theory, in which we have brought techniques from operator theory, operator algebras, and graph theory for the first time…
We study the design of graph filters to implement arbitrary linear transformations between graph signals. Graph filters can be represented by matrix polynomials of the graph-shift operator, which captures the structure of the graph and is…
A proof labelling scheme for a graph class $\mathcal{C}$ is an assignment of certificates to the vertices of any graph in the class $\mathcal{C}$, such that upon reading its certificate and the certificates of its neighbors, every vertex…
Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes…
We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that…
The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local…
Powerful techniques have been developed in quantum field theory that employ algebras of local operators, yet local operators cannot create physical charged states in gauge theory or physical nonzero-energy states in perturbative quantum…
For an $n \times n$ matrix $A$, let $q(A)$ be the number of distinct eigenvalues of $A$. If $G$ is a connected graph on $n$ vertices, let $\mathcal{S}(G)$ be the set of all real symmetric $n \times n$ matrices $A=[a_{ij}]$ such that for…
Locality-preserving logical operators in topological codes are naturally fault-tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure…
For any graph $G$ on $n$ vertices and for any {\em symmetric} subgraph $J$ of $K_{n,n}$, we construct an infinite sequence of graphs based on the pair $(G,J)$. The First graph in the sequence is $G$, then at each stage replacing every…
A recent development in graph-minor theory is to study local separators, vertex-sets that separate graphs locally but not necessarily globally. The local separators of a graph roughly correspond to the genuine separators of its local…
Mapping class groups of locally finite graphs are the analogue of those of infinite-type surfaces, and serve as a "big" version of $\text{Out}(F_n)$. In this paper, we investigate which of these mapping class groups have a dense conjugacy…
In the paper we introduce and study a classification of finite (simple, undirected, loopless) graphs with respect to a switch-equivalence (`local-complement' equivalence of \cite{pascvebl}, an analogue of the complement-equivalence of…
Storage codes on graphs are an instance of \emph{codes with locality}, which are used in distributed storage schemes to provide local repairability. Specifically, the nodes of the graph correspond to storage servers, and the neighbourhood…
Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists…
In [Phys. Rev. A 58, 1833 (1998)] a family of polynomial invariants which separate the orbits of multi-qubit density operators $\rho$ under the action of the local unitary group was presented. We consider this family of invariants for the…
The name graph state is used to describe a certain class of pure quantum state which models a physical structure on which one can perform measurement-based quantum computing, and which has a natural graphical description. We present the…
Graph isomorphism is an important computer science problem. The problem for the general case is unknown to be in polynomial time. The base algorithm for the general case works in quasi-polynomial time. The solutions in polynomial time for…
We study a problem motivated by a question related to quantum-error-correcting codes. Combinatorially, it involves the following graph parameter: $$f(G)=\min\set{|A|+|\{x\in V\setminus A : d_A(x)\text{is odd}\}| : A\neq\emptyset},$$ where…