Nonlocal-Adjacency Metric Dimension of Graphs

Let T = {t₁, t₂, . . ., t_k} ⊆ V (G) be an ordered subset of the vertex set of a graph G, and let u ∈ V (G) be a vertex in G. The adjacency metric representation of vertex u with respect to the set T is the k-vector r_A(u | T) = (d_A(u, t₁), d_A(u, t₂), . . ., d_A(u, t_k)). The set T is called a nonlocal-adjacency metric resolving set of the graph G if r_A(u | T) ≠ r_A(w | T) for every pair of vertices u, v ∈ G with u not adjacent to v. The minimum cardinality of a nonlocal-adjacency metric resolving set of G is called the nonlocal-adjacency metric dimension of G, denoted by dim_Anl(G). In this paper, we present graphs obtained from the degree corona product of two graphs. The degree corona product of graphs G and H, denoted by G ⊙_deg H, is the graph constructed by taking a graph G and (Formula presented) deg(vi) copies H_ij of graph H, and then connecting every vertex vi ∈ V (G) to all vertices in H_ij for every j ∈ {1, 2, . . ., deg(vi)} and i ∈ {1, 2, . . ., |V (G)|}. Furthermore, we determine and analyze the nonlocal-adjacency metric dimension of basic graphs G_b ∈ {P_n, C_n}, centered graphs G_c ∈ {K_n, S_n, K₁ + P_n, K₁ + C_n, K_m + K_n}, and the degree corona product graphs G_c ⊙_deg K₁. In addition, we provide upper bounds, characterizations of the nonlocal-adjacency metric dimension of graphs, and examples of applications of this concept.