On the metric dimension of amalgamation of sunflower and lollipop graph and caveman graph

Let G be a connected nontrivial graph with vertex set V ( G ) and edge set E ( G ). The distance between two vertices u and v in G is the shortest path length between u and v denoted d ( u, v ). Let W = { w 1 , w 2 , …, w k } be a subset of V ( G ). The representation of a vertex u with respect to W is a sequential pair of distances between u and all vertices in W , where u is a vertex in G . The set of W is called the resolving set if the representation of each vertex is different to W . Resolving set with a minimum cardinality called the metric basis and the number of element from some basis is called the metric dimension, denoted by dim ( G ). In this paper, we determine the metric dimension of amalgamation of sunfiower and lollipop graph ( SF n , v i ) ✱ ( L m, p , u p ) and caveman graph C ( n, m ). The results show that the metric dimension of amalgamation of sunflower and lollipop graph is dim (( SF n , v i ) ✱ ( L m, p , u p )) = m + 1 for n = 3, 4, …, 7; d i m ( ( S F n , v i ) * ( L m , p , u p ) ) = ⌈ n 3 ⌉ + m − 2 for n ≥ 8, and the metric dimension of caveman graph is dim ( C ( n, m )) = n for m = 3, 4; dim ( C ( n, m )) = ( m – 4) n for m ≥ 5.