Hasan Al-Ezeh

• PhD
• Professor (Full) at University of Jordan

Path graphs were proposed as a generalization of line graphs. The 2-path graph denoted by P2(G), of a graph G has vertex set the set of all paths of length two. Two such vertices are adjacent in the new graph if their union is a path of length three or a cycle of length three. In this paper we will introduce the path graph of the amalgamated graph...

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...

The geodetic, hull, and Steiner numbers of powers of paths

In this paper it is determined when the line graphs and the middle graphs of some classes of graphs are divisor graphs. Complete characterizations for cycles, trees, complete graphs and complete multipartite graphs whose line graphs (middle graphs) are divisor graphs are obtained. It is also shown that the line graphs and the middle graphs of the c...

A ring is called an almost pp-ring if the annihilator of each element of R is generated by its idempotents. We prove that for a ring R and an Abelian group G, if the group ring RG is an almost pp-ring then so is R, Moreover, if G is a finite Abelian group then |G|⁻¹∈R. Then we give a counter example to the converse of this. Also, we prove that RG i...

All rings R in this article are assumed to be commutative with unity 1 = 0. A ring R is called a GP F −ring if for every a ∈ R there exists a positive integer n such that the annihilator ideal Ann R (a n) is pure. We prove that for a ring R and an Abelian group G, if the group ring RG is a GP F −ring then so is R. Moreover, if G is a finite Abelian...

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 c be a proper k-coloring of a graph G. Let π={R1,R2,…,Rk} be the partition of V(G) induced by c, where Ri is the partition class receiving color i. The color code cπ(v) of a vertex v of G is the ordered k-tuple (d(v,R1),d(v,R2),…,d(v,Rk)), where d(v,Ri) is the minimum distance from v to each other vertex u∈Ri for 1≤i≤k. If all vertices of G hav...

Graph invariants are the properties of graphs that do not change under graph isomorphisms, the independent set decision problem, vertex covering problem, and matching number problem are known to be NP-Hard, and hence it is not believed that there are efficient algorithms for solving them. In this paper, the graph invariants matching number, vertex...

A graph G = (V(G),E(G)) is called 1-planar if it can be drawn in the plane such that every edge of the graph is cut by at most one other edge of the graph. For any ring R, the ideal intersection graph of R, denoted by G(R), is the graph whose vertices are the nontrivial proper ideals of R and two distinct vertices are adjacent if they have nontrivi...

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 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 .

We study regular cube-complementary graphs, that is, regular graphs whose complement and cube are isomorphic. We prove several necessary conditions for a graph to be regular cube-complementary, and characterize all cube-complementary circulant graphs with number of vertices is 9k where k is an integer.

This paper studies the unitary Cayley graph associated with ring of dual numbers, Z n [α]. It determines the exact diameter, vertex chromatic number and edge chromatic number. In addition, it classifies all perfect graphs within this class.

Suppose that T is a tree. AbuHijleh has shown that 2 T is a divisor graph iff T is a caterpillar with () 5 diam T . In this paper we characterize when 3 T and 4 T are divisor graphs.

We study cube-complementary graphs, that is, graphs whose complement and cube are isomorphic. We prove several necessary conditions for a graph to be cube-complementary, describe ways of building new cube-complementary graphs from existing ones, and construct infinite families of cube-complementary graphs.

In this article, we characterize for which finite commutative ring R, the zero-divisor graph Γ(R), the line graph L(Γ(R)) the complement graph Γ(R), and the line graph for the complement graph L(Γ(R)) are Eulerian.

In this paper, we show that Qnk is a divisor graph, for n = 2, 3. For n ≥ 4, we show that Qnk is a divisor graph iff k ≥ n-1. For folded-hypercube, we get FQn is a divisor graph when n is odd. But, if n ≥ 4 is even integer, then FQn is not a divisor graph. For n ≥ 5, we show that (FQn)k is not a divisor graph, where 2 ≤ k ≤ -n 2 ⌉-1.

Let Pn+1 be the path of order n + 1 on the vertices v0,v1,...,vn and Pkn+1 is the power of Pn+1. In this paper, we find the geodetic, hull, and Steiner numbers of Pkn+1.

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...

The zero-divisor graph of a commutative ring with unity (say R) is a graph whose vertices are the nonzero zero-divisors of this ring, where two distinct vertices are adjacent when their product is zero. This graph is denoted by Γ(R). In this paper, we study the structure of the zero-divisor graph Γ(ℤpn (x)) where p is an odd prime number, ℤpn is th...

In this paper, we prove that for any tree T, T2 is a divisor graph if and only if T is a caterpillar and the diameter of T is less than six. For any caterpillar T and a positive integer k ≥ 1 with diam(T) < 2k, we show that Tk is a divisor graph. Moreover, for a caterpillar T and k > 3 with diam(T) = 2k or diam(T) - 2k + 1, we show that Tk is a div...

The zero-divisor graph of a commutative ring with unity (say R) is a graph whose vertices are the nonzero zero-divisors of this ring, where two distinct vertices are adjacent when their product is zero. This graph is denoted by Gamma(R). In this paper, we study the structure of the zero-divisor graph Gamma(Z(pn) (x)) where p is an odd prime number,...

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...

We characterize the graphs G and H for which the Cartesian product G□H is a divisor graph. We show that divisor graphs form a proper subclass of perfect graphs. Also we prove that cycle permutation graphs of order at least 8 are divisor graphs if and only if they are perfect. Some results concerning amalgamation operations about obtaining new divis...

The zero-divisor graph of a commutative ring with one (say R) is a graph whose vertices are the nonzero zero-divisors of this ring, with two distinct vertices are adjacent in case their product is zero. This graph is denoted by Γ(R). We study the zero-divisor graph Γ(ℤ p n (α)), where p is a p prime number, ℤ p n is the set of integers modulo p n ,...

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...

It was conjectured in a recently published paper [the authors, Ars Comb. 94, 371–380 (2010; Zbl 1240.05163)] that for any integer k≥8 and any even integer n with 2k+3<n<2k+⌊k 2⌋+3, the k-th power C n k of the n-cycle is not a divisor graph. In this paper we prove this conjecture, hence obtaining a complete characterization of those powers of cycles...

In this paper, we study some operations which produce new divisor graphs from old ones. We prove that the contraction of a divisor graph along a bridge is a divisor graph. For two transmitters (receivers) u and v in some divisor orientation of a divisor graph G, it is shown that the merger G | u,v is also a divisor graph. Two special types of verte...

In this paper we prove that for any positive integers k,n with k≥2, the graph P n k is a divisor graph if and only if n≤2k+2, where P n k is the k th power of the path P n . For powers of cycles we show that C n k is a divisor graph when n≤2k+2, but is not a divisor graph when n≥2k+⌊k 2⌋+3, where C n k is the k th power of the cycle C n . Moreover,...

We provide a process to extend any bipartite diametrical graph of diameter 4 to an -graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets and , where , we prove that is a sharp upper bound of and construct an -graph in which this upper bound is attained, this graph can be viewed as a genera...

We presented a formula for the Wiener polynomial of the kth power graph. We use this formula to find the Wiener polynomials of the kth power graphs of paths, cycles, ladder graphs, and hypercubes. Also, we compute the Wiener indices of these graphs.

All rings considered in this paper are assumed to be commutative with identities. A ring R is a Q-ring if every ideal of R is a finite product of primary ideals. An almost Q-ring is a ring whose localization at every prime ideal is a Q-ring. In this paper, we first prove that the statements, R is an almost ZP I-ring and R(x) is an almost Q-ring are...

A diametrical graph G is said to be symmetric if d(u,v)+d(v,u¯)=d(G) for all u,v∈V(G), where u¯ is the buddy of u. If moreover, G is bipartite, then it is called an S-graph. It would be shown that the Cartesian product K2×C6 is not only the unique S-graph of order 12 and diameter 4, but also the unique symmetric diametrical graph of order 12 and di...

Let C(X) be the ring of all continuous real-valued functions defined on a completely regular T1-space. Let CΨ(X) and CK(X) be the ideal of functions with pseudocompact support and compact support, respectively. Further equivalent conditions are given to characterize when an ideal of C(X) is a P-ideal, a concept which was originally defined and char...

Let C(X) be the ring of all continuous real valued functions defined on a completely regular T1-space. Let CK(X) be the ideal of functions with compact support. Purity of CK(X) is studied and characterized through the subspace XL, the set of all points in X with compact neighborhoods (nbhd). It is proved that CK(X) is pure if and only if XL=∪f∈CK s...

Let L be a distributive lattice with 0 and 1, and let Spec L be the set of all proper prime ideals of L. Spec L can be endowed with two topologies, the spectral topology and D-topology. In this paper, it is proved that there is a bijection from the set of all σ-ideals of L to the set of all D-open subsets of Spec L. Let Max L and Min L be the sets...

Let R be a commutative ring with unity. In this paper, we prove that R is an almost PP–PM–ring if and only if R is an exchange PF–ring. Let X be a completely regular Hausdorff space, and let βX be the Stone Čech compactification of X. Then we prove that the ring C(X) of all continuous real valued functions on X is an almost PPÃ...

For a commutative ring with unity, A, it is proved that the power series ring A〚X〛 is a PF-ring if and only if for any two countable subsets S and T of A such that S⫅annA(T), there exists c∈annA(T) such that bc=b for all b∈S. Also it is proved that a power series ring A〚X〛 is a PP-ring if and only if A is a PP-rin...

For a commutative ring with unity R, it is proved that R is a PF-ring if and only if the annihilator, annR(a), for each a ϵ R is a pure ideal in R, Also it is proved that the polynomial ring, R[X], is a PF-ring if and only if R is a PF-ring. Finally, we prove that R is a PP-ring if and only if R[X] is a PP-ring.

Let C(X) be the ring of all continuous real valued functions defined on a completely regular T₁-space. Let C_K(X) be the ideal of functions with compact support. Purity of C_K(X) is studied and characterized through the subspace X_L, the set of all points in X with compact neighborhoods (nbhd). It is proved...