Related papers: Combinatorial Identities from the Spectral Theory …
We study a set of scattering matrices of quantum graphs containing minimal number of passbands, i.e., maximal number of zero elements. The cases of even and odd vertex degree are considered. Using a solution of inverse scattering problem,…
Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the…
In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing…
In this note, we find a combinatorial identity which is closely related to the multi-dimensional integral $\gamma_{m}$ in the study of divisor functions. As an application, we determine the finite dual of the group algebra of infinite…
A systematic procedure for generating certain identities involving elementary symmetric functions is proposed. These identities, as particular cases, lead to new identities for binomial and q-binomial coefficients.
In this paper, a new measurement to compare two large-scale graphs based on the theory of quantum probability is proposed. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. Our…
We analyze statistical properties of the complex system with conditions which manifests through specific constraints on the column/row sum of the matrix elements. The presence of additional constraints besides symmetry leads to new…
Vacillating tableaux are sequences of integer partitions that satisfy specific conditions. The concept of vacillating tableaux stems from the representation theory of the partition algebra and the combinatorial theory of crossings and…
This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements, about matrix models.
In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…
We introduce a hypergraph matrix, named the unified matrix, and use it to represent the hypergraph as a graph. We show that the unified matrix of a hypergraph is identical to the adjacency matrix of the associated graph. This enables us to…
We compute spectra of symmetric random matrices defined on graphs exhibiting a modular structure. Modules are initially introduced as fully connected sub-units of a graph. By contrast, inter-module connectivity is taken to be incomplete.…
Boolean combinations allow combining given combinatorial objects to obtain new, potentially more complicated, objects. In this paper, we initiate a systematic study of this idea applied to graphs. In order to understand expressive power and…
We describe recent advances in the study of random analogues of combinatorial theorems.
The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The…
Framed combinatorial topology is a novel theory describing combinatorial phenomena arising at the intersection of stratified topology, singularity theory, and higher algebra. The theory synthesizes elements of classical combinatorial…
We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.
We study some spectral properties of a matrix that is constructed as a combination of a Laplacian and an adjacency matrix of simple graphs. The matrix considered depends on a positive parameter, as such we consider the implications in…
From the standard procedure for constructing Feynman vacuum graphs in $\phi^4$ theory from the generating functional $Z$, we find a relation with sets of certain combinatorial matrices, which allows us to generate the set of all Feynman…
In the paper, with the aid of the series expansions of the square or cubic of the arcsine function, the authors establish several possibly new combinatorial identities containing the ratio of two central binomial coefficients which are…