ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II

Year 2020, , 61 - 74, 14.07.2020

Mohammad Abdulla Ayman Badawı

https://doi.org/10.24330/ieja.768135

Cited By: 1

Abstract

In 2015, the second-named author introduced the dot product
graph associated to a commutative ring $A$. Let $A$ be a
commutative ring with nonzero identity, $1 \leq n < \infty$ be
an integer, and $R = A \times A \times \cdots \times A$ ($n$
times). We recall that the total dot product graph of $R$
is the (undirected) graph $TD(R)$ with vertices $R^* = R\setminus
\{(0, 0, \dots, 0)\}$, and two distinct vertices $x$ and $y$ are
adjacent if and only if $x\cdot y = 0 \in A$ (where $x\cdot y$
denotes the normal dot product of $x$ and $y$). Let $Z(R)$ denote
the set of all zero-divisors of $R$. Then the zero-divisor
dot product graph of $R$ is the induced subgraph $ZD(R)$ of
$TD(R)$ with vertices $Z(R)^* = Z(R) \setminus \{(0, 0, \dots,
0)\}$. 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 $TD(R)$ with vertices $U(R)$. In this paper, we study
the structure of $TD(R)$, $UD(R)$, and $ZD(R)$ when $A = Z_n$ or
$A = GF(p^n)$, the finite field with $p^n$ elements, where $n \geq
2$ and $p$ is a prime positive integer.

Keywords

Dot product graph, annihilator graph, total graph, zero-divisor graph

References

• D. D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra, 159 (1993), 500-514.
• D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra, 36 (2008), 3073-3092.
• D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008), 2706-2719.
• D. F. Anderson and A. Badawi, On the total graph of a commutative ring without the zero element, J. Algebra Appl., 11(4) (2012), 1250074 (18 pp).
• D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl., 12(5) (2013), 1250212 (18 pp).
• D. F. Anderson and J. D. LaGrange, The semilattice of annihilator classes in a reduced commutative ring, Comm. Algebra, 43 (2015), 29-42.
• D. F. Anderson and J. D. LaGrange, Some remarks on the compressed zero-divisor graph, J. Algebra, 447 (2016), 297-321.
• D. F. Anderson and E. F. Lewis, A general theory of zero-divisor graphs over a commutative ring, Int. Electron. J. Algebra, 20 (2016), 111-135.
• D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999), 434-447.
• D. F. Anderson and S. B. Mulay, On the diameter and girth of a zero-divisor graph, J. Pure Appl. Algebra, 210 (2007), 543-550.
• D. F. Anderson and D. Weber, The zero-divisor graph of a commutative ring without identity, Int. Electron. J. Algebra, 23 (2018), 176-202.
• S. Akbari, H. R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra, 270 (2003), 169-180.
• S. Akbari, D. Kiani, F. Mohammadi and S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra, 213 (2009), 2224-2228.
• A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra, 42 (2014), 108-121.
• A. Badawi, On the dot product graph of a commutative ring, Comm. Algebra, 43 (2015), 43-50.
• I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988), 208-226.
• C. F. Kimball and J. D. LaGrange, The idempotent-divisor graphs of a commutative ring, Comm. Algebra, 46 (2018), 3899-3912.
• J. D. LaGrange, The x-divisor pseudographs of a commutative groupoid, Int. Electron. J. Algebra, 22 (2017), 62-77.
• W. J. LeVeque, Fundamentals of Number Theory, Addison-Wesley Publishing Company, Reading, Massachusetts, 1977.
• S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30 (2002), 3533-3558.
• R. Nikandish, M. J. Nikmehr and M. Bakhtyiari, Coloring of the annihilator graph of a commutative ring, J. Algebra Appl., 15(7) (2016), 1650124 (13 pp).
• Z. Pucanovic and Z. Petrovic, On the radius and the relation between the total graph of a commutative ring and its extensions, Publ. Inst. Math. (Beograd) (N.S.), 89 (2011), 1-9.
• P. K. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 (1995), 124-127.
• M. Sivagami and T. Tamizh Chelvam, On the trace graph of matrices, Acta Math. Hungar., 158 (2019), 235-250.
• S. Spiroff and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39 (2011), 2338-2348.
• T. Tamizh Chelvam and T. Asir, On the genus of the total graph of a commutative ring, Comm. Algebra, 41 (2013), 142-153.