# Sanhan KhasrawSalahaddin University - Erbil | SUH · Department of Mathematics

Sanhan Khasraw

PhD

## About

31

Publications

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## Publications

Publications (31)

A zero divisor graph of a finite ring is defined as a simple graph with its vertices are the zero divisors of the ring, and two distinct vertices are adjacent if and only if their product is equal to the zero element of the ring. In this research, the zero divisor graph is identified for the ring of 2x2 matrices over integers modulo prime. Since th...

Recently, the study of probabilities in ring theory has shown a significant increase in the field of algebra. Many interesting algebraic structures were modeled to find their probabilities in certain finite rings. In this paper, we introduce a new type of probability in finite rings, namely the squared-zero product probability. The aim is to study...

A graph is a mathematical subject that consists of vertices and edges. Many studies have been done on the graphs of algebraic structures, including groups and rings. A zero divisor graph of a finite ring is defined as a graph with all zero divisors of the ring as its vertices, where two vertices are adjacent when the product of the vertices is the...

Sombor index is a newly developed degree-based topological index which involves the degree of the vertex in a simple connected graph. The Sombor index is known as the square root of the sum of the squared degrees of two adjacent vertices in a graph. Meanwhile, the noncommuting graph associated to a group is a graph where its vertices are the non-ce...

In the field of algebra, the application of probability theory in ring theory has been widely studied by various researchers. In this paper, a type of probability in finite rings, namely the zero product probability is determined for some ring of 2x2 matrices with a single nonzero entry. To obtain the zero product probability, the exact order of th...

The study of rings and graphs has been explored extensively by researchers. To gain a more effective understanding on the concepts of the rings and graphs, more researches on graphs of different types of rings are required. This manuscript provides a different study on the concepts of commutative rings and undirected graphs. The non-zero divisor gr...

The zero-divisor graph of a ring is a graph whose vertices are the collection of zero-divisors of the ring, with two distinct vertices, x and y are connected by an edge if and only if xy=0. Meanwhile, a zero-divisor type graph is a compression of the zero-divisor graph by partitioning the vertices. For ring of integers modulo n, the zero-divisor ty...

In mathematics, mainly in the field of algebra, the study on probability related to groups and rings is a common topic which is widely discussed by many researchers. This study originated from the commutativity degree, which is introduced to find the probability that two elements in a group commute. Many extensions have been done on the commutativi...

For a finite group G, the intersection graph of G is the graph whose vertex set is the set of all proper non-trivial subgroups of G, where two distinct vertices are adjacent if their intersection is a non-trivial subgroup of G. In this article, we investigate the detour index, eccentric connectivity, and total eccentricity polynomials of the inters...

For a finite group G, the co-prime order graph Θ(G) of G is defined as the graph with vertex set G, the group itself, and two distinct vertices u, v in Θ(G) are adjacent if and only if gcd(o(u), o(v)) = 1 or a prime number. In this paper, some properties and some topological indices such as Wiener, Hyper-Wiener, first and second Zagreb, Schultz, Gu...

The study of graph theory was introduced and widely researched since many practical problems can be represented by graphs. A non-zero divisor graph is a graph in which its set of vertices is the non-zero elements of the ring and the vertices x and y are adjacent if and only if xy ≠ 0. In this study, we introduced the non-zero divisor graphs of some...

The study of graph properties has gathered many attentions in the past years. The graph properties that are commonly studied include the chromatic number, the clique number and the domination number of a finite graph. In this study, a type of graph properties, which is the perfect code is studied. The perfect code is originally used in coding theor...

Let R be a finite ring. The zero divisors of R are defined as two nonzero elements of R, say x and y where xy = 0. Meanwhile, the probability that two random elements in a group commute is called the commutativity degree of the group. Some generalizations of this concept have been done on various groups, but not in rings. In this study, a variant o...

Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is the identity of the group $G$. In this paper, we investigate some properties and exploring some topological in...

A non-commuting graph of a finite group $G$ is a graph whose vertices are non-central elements of $G$ and two vertices are adjacent if they don't commute in $G$. In this paper, we study the non-commuting graph of the group $U_{6n}$ and explore some of its properties including the independent number, clique and chromatic numbers. Also, the general f...

The study on probability theory in finite rings has been an interest of various researchers. One of the probabilities that has caught their attention is the probability that two elements of a ring have product zero. In this study, the probability is determined for a finite ring R of matrices over integers modulo four. First, the annihilators of R a...

For a nonabelian group G, the non-commuting graph $\Gamma_G$ of $G$ is defined as the graph with vertex set $G-Z(G)$, where $Z(G)$ is the center of $G$, and two distinct vertices of $\Gamma_G$ are adjacent if they do not commute in $G$. In this paper, we investigate the detour index, eccentric connectivity and total eccentricity polynomials of non-...

In this paper, the probability that two elements of a finite ring have product zero is considered. The bounds of this probability are found for an arbitrary finite commutative ring with identity 1. An explicit formula for this probability in the case of n Z , the ring of integers modulo n , is obtained.

Let G be a graph. G is said to be a zero divisor graph when its vertices, V(G) are all zero divisors of a finite ring R and two vertices are adjacent if and only if the product of the vertices is zero. In this study, the zero divisor graph is first constructed for the finite ring of square matrices over integers modulo two, using its definition and...

Let R be a finite ring. In this study, the probability that two random elements chosen from a finite ring have product zero is determined for some finite ring of matrices over Zn. Then, the results are used to construct the zero divisor graph which is defined as a graph whose vertices are the nonzero zero divisors of R and two distinct vertices x a...

Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and simplicity; and (2) sum decompositions.

An axial algebra is a commutative non-associative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion la...

Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and simplicity; and (2) sum decompositions.

An extension of the concept of commutativity degree named the probability that an element of a group fixes a set was introduced in 2013. Suppose is a metacyclic 5-group and is the set of all ordered pairs (x,y) in G*G such that lcm(|x|,|y|)=5, xy=yx and x is not equal to y. In this paper, the probability that an element of a metacyclic 5-group fixe...

Let G be a metacyclic 5-group and Ω is the set of all ordered pairs (x, y) in G × G such that lcm(|x|, |y|) = 5, xy = yx and x ≠ y. In this paper, the probability that an element of G fixes a set Ω is determined by using conjugation action. The results obtained are then applied to graph theory, more precisely to the orbit graph.

The main result of this thesis concerns the classification of 3-generated M-axial algebras A such that every 2-generated subalgebra of A is a Sakuma algebra of type N X, where N ∈ {2, 3, 4} and X ∈ {A, B, C}. This goal requires the classification of all groups G which are quotients of the groups T (s 1 ,s 2 ,s 3) = x, y, z | x 2 , y 2 , z 2 , (xy)...

This thesis is the classification of the Majorana algebras of the symmetric group S 4 of degree 4. There are twelve shapes in total for this group. Four of them are considered in [9]. We deal with six shapes not covered in [9], describe the resulting algebras and support our claims with hand and computer calculations. The two shapes not covered her...