
Manal GhanemUniversity of Jordan | UJ · Department of Mathematics
Manal Ghanem
Phd
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Publications (31)
Let G G be a graph. Then, the inverse graph G − 1 {G}^{-1} of G G is defined to be a graph that has adjacency matrix similar to the inverse of the adjacency matrix of G G , where the similarity matrix is ± 1 \pm 1 diagonal matrix. In this article, we introduced a generalization of this definition that serves the mixed graphs where the definition ap...
In this article we relate the six Prüfer conditions with the EM conditions. We use the EM-conditions to prove some cases of equivalence of the six Prüfer conditions. We also use the Prüfer conditions to answer some open problems concerning EM-rings.
The α-Hermitian adjacency matrix Hα of a mixed graph X has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number α. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as...
Let $G=(V,E)$ be a graph. The $k^{th}-$ power of $G$ denoted by $G^{k}$ is the graph whose vertex set is $V$ and in which two vertices are adjacent if and only if their distance in $G$ is at most $k.$ A vertex coloring of $G$ is acyclic if each bichromatic subgraph is a forest. A star coloring of $G$ is an acyclic coloring in which each bichromatic...
A mixed graph D is a graph that can be obtained from a graph by orienting some of its edges. Let α be a primitive n th root of unity, then the α−Hermitian adjacency matrix of a mixed graph is defined to be the matrix Hα = [hrs] where hrs = α if rs is an arc in D, hrs = α if sr is an arc in D, hrs = 1 if sr is a digon in D and hrs = 0 otherwise. In...
The $\alpha$-Hermitian adjacency matrix $H_\alpha$ of a mixed graph $X$ has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number $\alpha$. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization o...
Let [Formula: see text] be a commutative ring. A polynomial [Formula: see text] is an annihilating content (AC) polynomial if [Formula: see text] where [Formula: see text] and [Formula: see text] is a nonzerodivisor and [Formula: see text] is an EM-ring if each [Formula: see text] is an AC polynomial. In this paper, we investigate AC polynomials an...
Abstract. Let R be a commutative ring with unity. The total graph of R,
T (Γ(R)), is the simple graph with vertex set R and two distinct vertices x and
y are adjacent if x + y ∈ Z(R), where Z(R) is the set of all zero divisors of R. This
paper presents a study of some local properties of the graph T (Γ(Zn)). We answer
the question “ when is T (Γ(Zn...
This is a survey for all the work done so far on EM-rings, their extensions, and some of their generalizations.KeywordsPolynomial ringAnnihilating content polynomialEM-ring2010 Mathematics Subject Classification13A1513B2513E0513F2013F25
A ring R is called EM-Hermite if for each a, b ∈ R, there exist a 1 , b 1 , d ∈ R such that a = a 1 d, b = b 1 d and the ideal (a 1 , b 1) is regular. We give several characterizations of EM-Hermite rings analogue to those for K-Hermite rings, for example, R is an EM-Hermite ring if and only if any matrix in Mn,m(R) can be written as a product of a...
A commutative ring R with unityis called EM-Hermite if for each a, b ∈ R there exist c, d, f ∈ R such that a = cd, b = cf and the ideal (d, f) is regular in R. We showed in this article that R is a PP-ring if and only if the idealization R(+)R is an EM-Hermite ring if and only if R[x]/(x n+1) is an EM-Hermite ring for each n ∈ N. We generalize some...
A commutative ring R with unityis called EM-Hermite if for each a, b ∈ R there exist c, d, f ∈ R such that a = cd, b = cf and the ideal (d, f) is regular in R. We showed in this article that R is a PP-ring if and only if the idealization R(+)R is an EM-Hermite ring if and only if R[x]/(x n+1) is an EM-Hermite ring for each n ∈ N. We generalize some...
Let [Formula: see text] be the cycle graph of order [Formula: see text] on the vertices [Formula: see text] and [Formula: see text] be the [Formula: see text]th power of [Formula: see text]. In this paper, we find the hull number of [Formula: see text] under restricted conditions on the vertices of the graph [Formula: see text] namely the independe...
Let R be a commutative ring with unity.
Let c be a proper k-coloring of a graph G. Let π = { R 1 , R 2 , … , R k } be the partition of V ( G ) induced by c, where R i is the partition class receiving color i. The color code c π ( v ) of a vertex v of G is the ordered k-tuple ( d ( v , R 1 ) , d ( v , R 2 ) , … , d ( v , R k ) ) , where d ( v , R i ) is the minimum distance from v to each...
A ring R is called EM-Hermite if for each a, b ∈ R, there exist a 1 , b 1 , d ∈ R such that a = a 1 d, b = b 1 d and the ideal (a 1 , b 1) is regular. We give several characterizations of EM-Hermite rings analogue to those for K-Hermite rings, for example, R is an EM-Hermite ring if and only if any matrix in Mn,m(R) can be written as a product of a...
Let R be a commutative ring, G be an Abelian group, and let RG be the group ring. We say that RG is a U-group ring if a is a unit in RG if and only if (a) is a unit in R. We show that RG is a U-group ring if and only if G is a p-group and p ∈ J(R). We give some properties of U-group rings and investigate some properties of well known rings, such as...
Let R be a commutative ring with unity. The main objective of this article is to study the relationships between PP-rings, generalized morphic rings and EM-rings. Although PP-rings are included in the later rings, the converse is not in general true. We put necessary and sufficient conditions to ensure the converse using idealization and polynomial...
Let Cn be the cycle graph of order n on the vertices υ0, υ1;: : :, υn and Cnk be the k-th power of Cn. In this article we determine the hull-number of Cnk .
Let R be a commutative ring with unity. The total graph of R, T(Γ (R)), is the simple graph with vertex set R and two distinct vertices are adjacent if their sum is a zero-divisor in R. Let Reg(Γ(R)) and Z(Γ (R)) be the subgraphs of T(Γ (R)) induced by the set of all regular elements and the set of zero-divisors in R, respectively. We determine whe...
Let (R) be the zero divisor graph for a commutative ring with identity.
The k-domination number and the 2-packing number of (R), where R is an
Artinian ring, are computed. k-dominating sets and 2-packing sets for the
zero divisor graph of the ring of Gaussian integers modulo n, (Zn[i]), are
constructed. The center, the median, the core, as well...
All rings considered are commutative with unity and all groups considered are abelian. We give a characterization of a pure augmentation ideal, I(G), of a group ring, R(G). We study the relationship between the p-injectivity of R(G) and the p-injectivity of its ideal I(G). Keywords and phrases: Augmentation ideal, Pure ideal, P-injective ring, P-in...
Let R be any commutative ring with identity, and let C be a (finite or infinite) cyclic group. We show that the group ring R(C) is presimplifiable if and only if its augmentation ideal I(C) is presimplifiable. We conjecture that the group rings R(Cn) are presimplifiable if and only if n = p , p ∈ J(R), p is prime, and R is presimplifiable. We show...
Let Γ ( ℤ n [ i ]) be the zero divisor graph for the ring of the Gaussian integers modulo n . Several properties of the line graph of Γ ( ℤ n [ i ]) , L(Γ ( ℤ n [ i ])) are studied. It is determined when L(Γ ( ℤ n [ i ])) is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are...
The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally H and locally connected L(Γ(ℤ n [i]) ¯) is given. The chromatic number when n is a power of...
Keigher showed that quasi-prime ideals in differential commutative rings are analogues of prime ideals in commutative rings. In that direction, he introduced and studied new types of differential rings using quasi-prime ideals of a differential ring. In the same sprit, we define and study two new types of differential rings which lead to the mirror...
In this paper we study some properties of associate and presimplifiable rings. We give a characterization of the associate (resp., domainlike) pullback P of R1 → R3 ← R2 , where R1 and R2 are two presimplifiable (resp., domainlike) rings. We prove that R is presimplifiable ring if and only if the factor ring R/nil(R) is presimplifiable and the idea...
We use a translational invariant fuzzy subset p of a ring R to define two new types of commutative rings namely, p-presimplifiable and p-associate rings. We present some results of these rings. The interest of these results is that most of them are mirrors of corresponding results of presimplifiable and associate rings in classical ring theory.