Related papers: Zero sum partition into sets of the same order and…

We investigate eigenvalues of the zero-divisor graph $\Gamma(R)$ of finite commutative rings $R$ and study the interplay between these eigenvalues, the ring-theoretic properties of $R$ and the graph-theoretic properties of $\Gamma(R)$. The…

Hefetz, M\"{u}tze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper we support the analogous question for distance magic labeling. Let $\Gamma$ be an Abelian group of order $n$. A…

Let $R$ be a ring with involution $*$ and $Z^*(R)$ denotes the set of all non-zero zero-divisors of $R$. We associate a simple (undirected) graph $\Gamma'(R)$ with vertex set $Z^*(R)$ and two distinct vertices $x$ and $y$ are adjacent in…

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and…

A \emph{group distance magic labeling} or a $\gr$-distance magic labeling of a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian group $\gr$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every…

We introduce the vertex-arboricity of group-labelled graphs. For an abelian group $\Gamma$, a $\Gamma$-labelled graph is a graph whose edges are labelled by elements of $\Gamma$. For an abelian group $\Gamma$ and $A\subseteq \Gamma$, the…

The Abels-Margulis-Soifer lemma states that if a semigroup $\Gamma$ acts strongly irreducibly by linear transformations on a finite-dimensional real vector space, then any element of $\Gamma$ can be multiplied by an element of some fixed…

For a commutative semigroup $S$ with 0, the zero-divisor graph of $S$ denoted by $\Gamma(S)$ is the graph whose vertices are nonzero zero-divisor of $S$, and two vertices $x$, $y$ are adjacent in case $xy=0$ in $S$. In this paper we study…

Let $\Gamma$ be a countable abelian group. An (abstract) $\Gamma$-system $\mathrm{X}$ - that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $\Gamma$ - is said to be a Conze-Lesigne system if…

We analyze orbifolds with discrete torsion of the ABJM theory by a finite subgroup $\Gamma$ of $SU(2)\times SU(2)$ . Discrete torsion is implemented by twisting the crossed product algebra resulting after orbifolding. It is shown that, in…

In this paper we study sub-semigroups of a zero-divisor semigroup $S$ determined by properties of the zero-divisor graph $\Gamma(S)$. We use these sub-semigroups to study the correspondence between zero- divisor semigroups and zero-divisor…

Let $K$ be a global function field of characteristic $p$, and let $\Gamma$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $\Gamma$ is a $p$-torsion element. We define…

For a finite (not necessarily Abelian) group $(\Gamma,\cdot)$, let $n(\Gamma) \in \mathbb{N}$ denote the smallest positive integer $n$ such that for every labelling of the arcs of the complete digraph of order $n$ using elements from…

Absence of (complex) zeros property is at the heart of the interpolation method developed by Barvinok \cite{barvinok2017combinatorics} for designing deterministic approximation algorithms for various graph counting and computing partition…

Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p. The countable Abelian p-groups are classified by Ulm's theorem, and Khisamiev…

The distribution of cardinalities of zero-sum sets in abelian groups is completely determined. A complex summation involving the M\"obius function is given for the general abelian group, while in many special cases, including the case of…

It is shown that images of cross-sections of surjective morphisms $f: \Gamma \longrightarrow \Delta$ of divisible abelian $o$-groups are exactly divisible, tame (equivalently, relative Dedekind complete) and cofinal subgroups of $\Gamma$…

For an additive group $\Gamma$ the sequence $S = (g_1, \ldots, g_t)$ of elements of $\Gamma$ is a zero-sum sequence if $g_1 + \cdots + g_t = 0_\Gamma$. The cross number of $S$ is defined to be the sum $\sum_{i=1}^k 1/|g_i|$, where $|g_i|$…

We prove a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs $(G,\gamma)$ where $\gamma$ assigns to each edge of an undirected graph $G$ an element of an abelian group $\Gamma$. As a…

For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable…