On the dot product graph of a commutative ring II
View/
Open
Date
2020Author
Abdulla, Mohammad Ahmad
Badawi, Ayman
Advisor(s)
Unknown advisorType
PeerReviewed
Article
Published version
Metadata
Show full item recordAbstract
In 2015, the secondnamed author introduced the dot product graph associated to a commutative ring A. Let A be a commutative ring with nonzero identity, 1 ≤ n < ∞ be an integer, and R = A × A × · · · × A (n times). We recall that the total dot product graph of R is the (undirected) graph T D(R) with vertices R∗ = R \ {(0, 0, . . . , 0)}, and two distinct vertices x and y are adjacent if and only if x·y = 0 ∈ A (where x·y denotes the normal dot product of x and y). Let Z(R) denote the set of all zerodivisors of R. Then the zerodivisor dot product graph of R is the induced subgraph ZD(R) of T D(R) with vertices Z(R)∗ = Z(R) \ {(0, 0, . . . , 0)}. Let U(R) denote the set of all units of R. Then the unit dot product graph of R is the induced subgraph UD(R) of T D(R) with vertices U(R). In this paper, we study the structure of T D(R), UD(R), and ZD(R) when A = Zn or A = GF(pn), the finite field with pn elements, where n ≥ 2 and p is a prime positive integer.DSpace URI
http://hdl.handle.net/11073/18297External URI
https://doi.org/10.24330/ieja.76813513066048Collections
Related items
Showing items related by title, author, creator and subject.

On The Unit Dot Product Graph Of A Commutative Ring.
Abdulla, Mohammad Ahmad (201601)In 2015, Ayman Badawi (Badawi, 2015) introduced the dot product graph associated to a commutative ring A. Let A be a commutative ring with nonzero identity, 1 n < 1 be an integer, and R =A x A x ... x A (n times). We ... 
Graph of Linear Transformations over a Field
ElAshi, Yasmine Ahmed (201906) 
Brain Connectivity Analysis Under Semantic Vigilance and Enhanced Mental States
Yahya, Fares; Tariq, Usman; Hassanin, Omnia; Mir, Hasan; Babiloni, Fabio; AlNashash, Hasan (MDPI, 2019)In this paper, we present a method to quantify the coupling between brain regions under vigilance and enhanced mental states by utilizing partial directed coherence (PDC) and graph theory analysis (GTA). The vigilance state ...