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On fractional metric dimension of comb product graphs

Abstract:

A vertex z in a connected graph G resolves two vertices u and v in G if dG(u, z) = dG(v, z). A set of vertices RG(u, v} is a set of all resolving vertices of u and v in G. For every two distinct vertices u and v in G, a resolving function f of G is a real function f: V (G) → [0; 1] such that f(RG(u, v)) ≥ 1. The minimum value of f(V (G)) from all resolving functions f of G is called the fractional metric dimension of G. In this paper, we consider a graph which is obtained by the comb product between two connected graphs G and H, denoted by Go H. For any connected graphs G, we determine the fractional metric dimension of Go H where H is a connected graph having a stem or a major vertex.