Related papers: Frames for Graph Signals on the Symmetric Group: A…
For any set representation (permutation representation) of the symmetric group $S_n$, we give combinatorial interpretation for coefficients of its Frobenius character expanded in the basis of monomial symmetric functions.
In this paper we introduce a Cayley-type graph for group-subgroup pairs and present some elementary properties of such graphs, including connectedness, their degree and partition structure, and vertex-transitivity. We relate these…
Owing to their interesting spectral properties, the synthetic crystals over lattices other than regular Euclidean lattices, such as hyperbolic and fractal ones, have attracted renewed attention, especially from materials and meta-materials…
A Cayley Graph for a group $G$ is called normal edge-transitive if it admits an edge-transitive action of some subgroup of the Holomorph of $G$ (the normaliser of a regular copy of $G$ in $\operatorname{Sym}(G)$). We complete the…
Let $\Gamma$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the…
For a set of quantum states generated by the action of a group, we consider the graph obtained by considering two group elements adjacent whenever the corresponding states are non-orthogonal. We analyze the structure of the connected…
These lecture notes are on automorphism groups of Cayley graphs and their applications to optimal fault-tolerance of some interconnection networks. We first give an introduction to automorphisms of graphs and an introduction to Cayley…
We present a framework for representing and modeling data on graphs. Based on this framework, we study three typical classes of graph signals: smooth graph signals, piecewise-constant graph signals, and piecewise-smooth graph signals. For…
A group $G$ has a Frobenius graphical representation (GFR) if there is a simple graph $\varGamma$ whose full automorphism group is isomorphic to $G$ and it acts on vertices as a Frobenius group. In particular, any group $G$ with GFR is a…
A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of $\mathrm{PSL}_{n}(q)$…
Graph signal processing analyzes signals supported on the nodes of a graph by defining the shift operator in terms of a matrix, such as the graph adjacency matrix or Laplacian matrix, related to the structure of the graph. With respect to…
According to the O'Nan--Scott Theorem, a finite primitive permutation group either preserves a structure of one of three types (affine space, Cartesian lattice, or diagonal semilattice), or is almost simple. However, diagonal groups are a…
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or…
We investigate subsets of the symmetric group with structure similar to that of a graph. The trees of these subsets correspond to minimal conjugate generating sets of the symmetric group. There are two main theorems in this paper. The first…
We consider the problem of designing spectral graph filters for the construction of dictionaries of atoms that can be used to efficiently represent signals residing on weighted graphs. While the filters used in previous spectral graph…
We introduce and study a family of inhomogeneous symmetric functions which we call the Frobenius-Schur functions. These functions are indexed by partitions and differ from the conventional Schur functions in lower terms only. Our interest…
We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This…
The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In…
Let $S$ be a set of transpositions that generates the symmetric group $S_n$, where $n \ge 3$. The transposition graph $T(S)$ is defined to be the graph with vertex set $\{1,\ldots,n\}$ and with vertices $i$ and $j$ being adjacent in $T(S)$…
The semidirect product of a finitely generated group dual with the symmetric group can be described through so-called group-theoretical categories of partitions (covers only a special case; due to Raum--Weber, 2015) and skew categories of…