Dot product graphs and domination number

Authors

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

Abstract

Let A be a commutative ring with 1 = 0 and R = A × A. The unit dot product graph of
R is defined to be the undirected graph UD(R) with the multiplicative group of units in
R, denoted by U(R), as its vertex set. 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. In 2016,
Abdulla studied this graph when A = Zn, n ∈ N, n ≥ 2. Inspired by this idea, we study
this graph when A has a finite multiplicative group of units. We define the congruence
unit dot product graph of R to be the undirected graph CUD(R) with the congruent
classes of the relation ∼ defined on R as its vertices. Also, we study the domination
number of the total dot product graph of the ring R = Zn × ... × Zn, k times and
k < ∞, where all elements of the ring are vertices and adjacency of two distinct
vertices is the same as in UD(R). We find an upper bound of the domination number of
this graph improving that found by Abdulla.

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