相关论文: On the Ingleton-Violations in Finite Groups

A fundamental problem in mathematics and network analysis is to find conditions under which a graph can be partitioned into smaller pieces. The most important tool for this partitioning is the Fiedler vector or discrete Cheeger inequality.…

For a finite group $G$, the size of a minimum generating set of $G$ is denoted by $d(G)$. Given a finite group $G$ and an integer $k$, deciding if $d(G)\leq k$ is known as the minimum generating set (MIN-GEN) problem. A group $G$ of order…

Classifying orthogonal arrays is a well known important class of problems that asks for finding all non-isomorphic, non-negative integer solutions to a class of systems of constraints. Solved instances are scarce. We develop two new methods…

We consider the group isomorphism problem: given two finite groups G and H specified by their multiplication tables, decide if G and H are isomorphic. The n^(log n) barrier for group isomorphism has withstood all attacks --- even for the…

The \textit{boxicity} (\textit{cubicity}) of an undirected graph $\Gamma$ is the smallest non-negative integer $k$ such that $\Gamma$ can be represented as the intersection graph of axis-parallel rectangular boxes (unit cubes) in…

Complex networks contain complete subgraphs such as nodes, edges, triangles, etc., referred to as simplices and cliques of different orders. Notably, cavities consisting of higher-order cliques play an important role in brain functions.…

In this paper we study some open questions related to the smallest order $f({\cal C},\lnot {\cal H})$ of a 4-regular graph which has a connectivity property ${{\cal C}}$ but does not have a hamiltonian property ${\cal H}$. In particular,…

For a finite group $G$, the prime graph $\Gamma(G)$ (also known as Gruenberg-Kegel graph) is defined to be the graph where the vertices are the primes that divide $|G|$ such that two vertices $p$ and $q$ share an edge if and only if there…

This paper investigates topological indices for the greedy tree $\mathcal{T}_\mathscr{D}$ associated with a graphic degree sequence $\mathscr{D} = (d_1 \geqslant d_2 \geqslant \dots \geqslant d_n)$ of a tree. A fundamental challenge in the…

Given n (discrete or continuous) random variables X_i, the (2^n-1)-dimensional vector obtained by evaluating the joint entropy of all non-empty subsets of {X_1,...,X_n} is called an entropic vector. Determining the region of entropic…

The classification of the finite subgroups of $\mathrm{GL}_n(\mathbb{C})$ and $\mathrm{PGL}_n(\mathbb{C})$ is a classical problem in the field of finite group theory, dating back to the late 19th century with authors like Klein, Jordan,…

We make progress in the following three problems: 1. Constructing optimal seeded non-malleable extractors; 2. Constructing optimal privacy amplification protocols with an active adversary, for any security parameter; 3. Constructing…

In two recent papers (Phys. Rev. A90 (2014), 062121 and Phys. Rev. A91 (2015), 052110) an interesting method of analyzing the violation of Bell inequalities has been proposed which is based on the theory of finite group representations. We…

Group network codes are a generalization of linear codes that have seen several studies over the last decade. When studying network codes, operations performed at internal network nodes called local encoding functions, are of significant…

We investigate the \textit{group irregularity strength}, $s_g(G)$, of a graph, i.e. the least integer $k$ such that taking any Abelian group $\mathcal{G}$ of order $k$, there exists a function $f:E(G)\rightarrow \mathcal{G}$ so that the…

We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of…

Given a group $G$ and an integer $n \geq 0$, let $\mathcal{F}_n$ denote the family of all virtually abelian subgroups of $G$ of rank at most $n$. In this article, we show that for each $n \geq 1$, the minimal dimension of a model for the…

Given a set of vertices in a network, that we believe are of interest for the application under analysis, community search is the problem of producing a subgraph potentially explaining the relationships existing among the vertices of…

For a (possibly infinite) fixed family of graphs F, we say that a graph G overlays F on a hypergraph H if V(H) is equal to V(G) and the subgraph of G induced by every hyperedge of H contains some member of F as a spanning subgraph.While it…

We consider the problem of designing succinct data structures for interval graphs with $n$ vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time in the $\Theta(\log n)$-bit word RAM model. The…