Extended dot product graph of a commutative ring.

Let B be a commutative ring with 1 ≠ 0 , 1 ≤ m < ∞ be an integer and R = B × B × ⋯ × B (m times). The extended total dot product graph TD ¯ (R) and extended zero-divisor dot product graph ZD ¯ (R) are undirected graphs with vertex sets R ∗ = R \ { (0 , 0 , ... 0) } and Z ∗ (R) = Z (R) \ { (0 , 0 , ... 0) } respectively. Two distinct vertices a = (a 1 , a 2 , ... , a m) and b = (b 1 , b 2 , ... , b m) are adjacent in TD ¯ (R) and ZD ¯ (R) if there exist positive integers k and ℓ such that a k · b ℓ = 0 with a k ≠ 0 and b ℓ ≠ 0 respectively (where a k · b ℓ = a 1 k b 1 ℓ + a 2 k b 2 ℓ + · · · + a m k b m ℓ ∈ B , denotes the normal dot product of a k and b ℓ ). In this paper, we study about connectedness, diameter and girth of TD ¯ (R) and ZD ¯ (R) . We also characterize all rings R for which TD ¯ (R) and ZD ¯ (R) are planar and outerplanar. [ABSTRACT FROM AUTHOR]