Study on fractional metric dimensions of connected graphs
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Date
2025-08-07
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UMT Lahore
Abstract
Consider a graph N = (V (N), E(N)), where V (N) represents the set of vertices and E(N) ⊆ V (N) × V (N) represents the set of edges. A walk in N is defined as a sequence of alternating vertices and edges. A path is a specific type of walk in which no vertices are repeated, except possibly the first and last vertices. For any graph N = (V (N), E(N)), the distance d(u, v) between two vertices u and v is the length of the shortest path connecting them. A graph N is classified as connected if a path exists between every pair of vertices. For a vertex u ∈ V (N), the degree of u, denoted as d(u), refers to the number of vertices directly adjacent to u. The neighborhood of u is defined as N(u) = {v ∈ V (N) : uv ∈ E(N)}, and the second neighborhood set of u, denoted as N2(u), is given by N2(u) = {v ∈ V (N) : d(u, v) = 2}. The degree sequence of N is the ordered list of vertex degrees in non-increasing order. The independence number α(N) of N is the size of the largest set of vertices in N such that no two vertices in the set are adjacent. Let U = {u1, u2, u3, . . . , um} ⊆ V (N), then a m-tuple metric form of u ∈ V (N) with respect to U is given by r(u|U ) = (d(u, u1), d(u, u2), . . . , d(u, um)). The set U is called a resolving set if every pair of distinct vertices in N has unique metric representation. The smallest resolving set, in term of cardinality, is named as metric basis of N and its cardinality is referred as the metric dimension of N. A vertex w is said to resolve a pair of vertices {u, v} if d(w, u) ̸ = d(w, v). A resolving neighborhood set for a pair of vertices {u, v} ⊆ V (N), denoted by R(u, v), is the set of all vertices that resolve the pair {u, v}, i.e., R(u, v) = {w ∈ V (N) | d(w, u) ̸= d(w, v)}. A function µ : V (N) → [0, 1] is called a resolving function if, for every resolving neighborhood set R(u, v), µ(R(u, v)) = ∑{w∈R(u,v)} µ(w) ≥ 1. A resolving function µ is termed a minimal resolving function if, for any other function φ : V (N) → [0, 1] such that φ ≤ µ and φ (w) ̸ = µ(w) for at least one w ∈ V (N), φ is not the resolving function of N. The fractional metric dimension of a graph N is defined as dimFm(N) = η, where η= min{|µ| : µ is minimal resolving function of N}.