let $mathcal {n}_g$ denote the set of all proper‎ ‎normal subgroups of a group $g$ and $a$ be an element of $mathcal‎ ‎{n}_g$‎. ‎we use the notation $ncc(a)$ to denote the number of‎ ‎distinct $g$-conjugacy classes contained in $a$ and also $mathcal‎ ‎{k}_g$ for the set ${ncc(a) | ain mathcal {n}_g}$‎. ‎let $x$ be‎ ‎a non-empty set of positive integers‎. ‎a group $g$ is said to be‎ ‎$x$-d...

An H-magic labeling in a H-decomposable graph G is a bijection f : V (G) ∪ E(G) → {1, 2, ..., p + q} such that for every copy H in the decomposition, ΣνεV(H) f(v) +  ΣeεE(H) f(e) is constant. f is said to be H-E-super magic if f(E(G)) = {1, 2, · · · , q}. A family of subgraphs H1,H2, · · · ,Hh of G is a mixed cycle-decomposition of G if every subgraph Hi is isomorphic to some cycle Ck, for k ≥ ...

An H-magic labeling in a H-decomposable graph G is a bijection f : V (G) ∪ E(G) → {1, 2, ..., p + q} such that for every copy H in the decomposition, ∑νεV (H) f(v) + ∑νεE (H) f(e) is constant. f is said to be H-E-super magic if f(E(G)) = {1, 2, · · · , q}. A family of subgraphs H1,H2, · · · ,Hh of G is a mixed cycle-decomposition of G if every subgraph Hi is isomorphic to some cycle Ck, for k ≥...

It is well-known that all orthogonal arrays of the form OA(N, t+1, 2, t) are decomposable into λ orthogonal arrays of strength t and index 1. While the same is not generally true when s = 3, we will show that all simple orthogonal arrays of the form OA(N, t + 1, 3, t) are also decomposable into orthogonal arrays of strength t and index 1.

Let $mathcal {N}_G$ denote the set of all proper‎ ‎normal subgroups of a group $G$ and $A$ be an element of $mathcal‎ ‎{N}_G$‎. ‎We use the notation $ncc(A)$ to denote the number of‎ ‎distinct $G$-conjugacy classes contained in $A$ and also $mathcal‎ ‎{K}_G$ for the set ${ncc(A) | Ain mathcal {N}_G}$‎. ‎Let $X$ be‎ ‎a non-empty set of positive integers‎. ‎A group $G$ is said to be‎ ‎$X$-d...

‎Recently‎, ‎some techniques such as adding whiskers and attaching graphs to vertices of a given graph‎, ‎have been proposed for constructing a new vertex decomposable graph‎. ‎In this paper‎, ‎we present a new method for constructing vertex decomposable graphs‎. ‎Then we use this construction to generalize the result due to Cook and Nagel‎.

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of mu...

During the last decades several learning algorithms have been proposed to learn probability distributions based on decomposable models. Some of these algorithms can be used to search for a maximum likelihood decomposable model with a given maximum clique size, k. Unfortunately, the problem of learning a maximum likelihood decomposable model given a maximum clique size is NP-hard for k > 2. In t...

Let C be a curve of genus 2 and ψ1 : C −→ E1 a map of degree n, from C to an elliptic curve E1, both curves defined over C. This map induces a degree n map φ1 : P 1 −→ P 1 which we call a Frey-Kani covering. We determine all possible ramifications for φ1. If ψ1 : C −→ E1 is maximal then there exists a maximal map ψ2 : C −→ E2, of degree n, to some elliptic curve E2 such that there is an isogeny...

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraphK n is decomposable into commuting perfect matchings if and only if n is a 2-power. Also, it is shown that the complete graph Kn is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 200...