Research Article
Year 2020, Volume 28, Issue 28, 61 - 74, 14.07.2020

### 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.

### ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II

Year 2020, Volume 28, Issue 28, 61 - 74, 14.07.2020

### 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.

### 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.

### Details

Primary Language English Mathematics Articles Mohammad ABDULLA This is me The American University of Sharjah United Arab Emirates Ayman BADAWI This is me (Primary Author) The American University of Sharjah United Arab Emirates July 14, 2020 Year 2020, Volume 28, Issue 28