Related papers: Signed Difference Sets

Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We…

A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ does not equal the sum of $l$ (not necessarily distinct) elements of $A$. We are interested in finding…

Let $F_n$ be a free group of rank $n$, with free generating set $X$. A subset $D$ of $F_n$ is a \emph{Distinct Difference Configuration} if the differences $g^{-1}h$ are distinct, where $g$ and $h$ range over all (ordered) pairs of distinct…

For a finite abelian group $G$ written additively, and a non-empty subset $A\subset [1,\exp(G)-1]$ the weighted Davenport Constant of $G$ with respect to the set $A$, denoted $D_A(G)$, is the least positive integer $k$ for which the…

The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$…

A signed dominating function of graph $\Gamma$ is a function $g :V(\Gamma) \longrightarrow \{-1,1\}$ such that $\sum_{u \in N[v]}g(u) >0$ for each $v \in V(\Gamma)$. The signed domination number $\gamma_{_S}(\Gamma)$ is the minimum weight…

A set of permutations is called sign-balanced if the set contains the same number of even permutations as odd permutations. Let $S_n(\sigma_1, \sigma_2, \ldots, \sigma_r)$ be the set of permutations in the symmetric group $S_n$ which avoids…

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order…

For a finite abelian group $G$ and a splitting field $K$ of $G$, let $d(G, K)$ denote the largest integer $l \in \N$ for which there is a sequence $S = g_1 \cdot ... \cdot g_l$ over $G$ such that $(X^{g_1} - a_1) \cdot ... \cdot (X^{g_l} -…

A difference set is said to have classical parameters if $ (v,k, \lambda) = (\frac{q^d-1}{q-1}, \frac{q^{d-1}-1}{q-1}, \frac{q^{d-2}-1}{q-1}).$ The case $d=3$ corresponds to planar difference sets. We focus here on the family of abelian…

Let R be a finite commutative ring with unity, and let G = (V, E) be a simple graph. The zero-divisor graph, denoted by {\Gamma}(R) is a simple graph with vertex set as R, and two vertices x, y \in R are adjacent in {\Gamma}(R) if and only…

Given an integer $k\ge3$ and a group $G$ of odd order, if there exists a $2$-$(v,k,1)$-design and if $v$ is sufficiently large, then there is such a design whose automorphism group has a subgroup isomorphic to $G$. A weaker result is proved…

A higher order difference equation may be generally defined in an arbitrary nonempty set S as: \[ f_{n}(x_{n},x_{n-1},...,x_{n-k})=g_{n}(x_{n},x_{n-1},...,x_{n-k}) \] where $f_{n},g_{n} :S^{k+1}\rightarrow S$ are given functions for…

Let $G$ be a finite group. For all $a \in \Z$, such that $(a,|G|)=1$, the function $\rho_a: G \to G$ sending $g$ to $g^a$ defines a permutation of the elements of $G$. Motivated by a recent generalization of Zolotarev's proof of classic…

A subset $B$ of an Abelian group $G$ is called a difference basis of $G$ if each element $g\in G$ can be written as the difference $g=a-b$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is…

A signed graph is a pair $(G,\Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V={1,...,n}$ and $\Sigma\subseteq E$. By $S(G,\Sigma)$ we denote the set of all symmetric $V\times V$…

The representation ring of an affine algebraic group scheme can be endowed with the structure of a (special) $\lambda$-ring. We show that the same is true for the ring of symmetric representations, i.e. for the Grothendieck-Witt ring of the…

A signed graph is an ordered pair $\Sigma=(G,\sigma),$ where $G=(V,E)$ is the underlying graph of $\Sigma$ with a signature function $\sigma:E\rightarrow \{1,-1\}$. In this article, we define $n^{th}$ power of a signed graph and discuss…

A signed graph is a graph where the edges are assigned labels of either "$+$" or "$-$". The sign of a cycle in the graph is the product of the signs of its edges. We equip each signed complete graph with a vector whose entries are the…

A signed graph is a graph with edges marked positive and negative; it is unbalanced if some cycle has negative sign product. We introduce the concept of vector valued switching function in signed graphs, which extends the concept of…