On local adjacency metric dimension of some wheel related graphs with pendant points

Abstract

Let G = (V(G),E(G)) be any connected graph of order n = |V(G)| and measure m = |E(G)|. For an order set of vertices S = { s1, s2, …, sk} and a vertex v in G, the adjacency representation of v with respect to S is the ordered k-tuple rA(v|S) = (dA(v, s1), dA(v, s2), …, dA(v, sk)), where dA(u,v) represents the adjacency distance between the vertices u and v. The set S is called a local adjacency resolving set of G if for every two distinct vertices u and v in G, u adjacent v then rA(u|S)≠rA(v|S). A minimum local adjacency resolving set for G is a local adjacency metric basis of G. Local adjacency metric dimension for G, dimA,l(G), is the cardinality of vertices in a local adjacency metric basis for G. In this paper, we study and determine the local adjacency metric dimension of some wheel related graphs G (namely gear graph, helm, sunflower and friendship graph) with pendant points, that is edge corona product of G and a trivial graph K1, G◊K1. Moreover, we compare among the local adjacency metric dimension of G◊K1 graph,of Wn◊K1 graph and metric dimension of Wn.