On Commutative Characterization of Graph Operation with Respect to Metric Dimension

Abstract

Let be a connected graph with vertex set and

,

,…, ⊆ . A representation of a vertex ∈) with respect to is an ordered m-tuple |

,

,

,

,...,

,

where , is the distance between vertices and . The set is called a resolving set for if every vertex of has a distinct representation with respect to W. A resolving set containing a minimum number of vertices is called a basis for . The metric dimension of , denoted by dim , is the number of vertices in a basis of . In general, the comb product and the corona product are noncommutative

operations in a graph. However, these operations can be commutative with respect to the metric dimension for some graphs with certain conditions. In this paper, we determine the metric dimension of the generalized comb and corona products of graphs and the necessary and sufficient conditions of the graphs in order for the comb and corona products to be commutative operations with respect to the metric dimension.