The mixed metric dimension of wheel-like graphs

Abstract

Consider the graph G = (V, E). It is a connected graph. It is a simple graph too. A node w ∈ V, then we call vertex, determined two elements of graph. There are vertices and edges of graphs. Any two vertices x, y ∈ E ∪ V if d(w, x) ≠ d(w, y), which d(w, x) and d(w, y) is the mixed distance of the element w (vertices or edges) in graph G. A set of vertices in a graph G is represented by the symbol Rm that defines a mixed metric generator for G, if the elements of vertices or edges are stipulated by several vertex set of Rm . There’s a chance that some mixed metric generators have varied cardinality. We choose one whose the minimum cardinality and it is called the mixed metric dimension of graph G, denoted by dimm (G). This research examines the mixed metric dimension of gear Gn , helm Hn , sunflower SFn , and friendship graph Frn . We call these graphs by wheel-like graphs. Our findings include the mixed metric dimension of gear graph Gn of order n ≥ 4 is dimm (Gn ) = n, helm graph Hn of order n ≥ 4 is dimm (Hn ) = n, sunflower graph SFn of order n ≥ 5 is dimm (SFn ) = n and friendship graph Frn of order n ≥ 3 is dimm (Frn ) = 2n.