Various types of basic graphs , which are commonly utilized in the construction of new graphs through graph operations, are described in this part. The definitions in this subsection are referenced from [16].

Definition 2.1.[16]A path graph , is a graph with order n and size . The set of vertices in is where and the set of edges

Definition 2.2.[16]A cycle graph is a graph with order n and size where , with the vertex set and the edge set

Definition 2.3.[17]A star graph is a tree consisting of n vertices, in which a single vertex has degree , while each of the remaining vertices has degree one.

Definition 2.4.[16]A complete graph is a graph that has n vertices, where every vertex forms an edge with each other vertex. A Complete graph has edges.

When , the complete graph is referred to a trivial graph, and is also known as a null graph or an empty graph.

In this part, the corona and joint product between graphs G and H are discussed. The corona operation was originally introduced in [4], and its construction can be found in Definition 2.5. The joint operation was introduced in [17] and described in Definition 2.6.

Definition 2.5.[4]Let G and H be two graphs. The corona product of G and H, denoted , is a graph obtained by taking one copy of G and creating copies of H, denoted as for each , and then joininh every vertex in to the vertex of G. The resulting graph is referred to the corona graph.

Figure 5 illustrates the graph obtained from the corona operation between , whose vertices are colored red, and , whose vertices are colored blue.

Definition 2.6.[18]The joint operation of G and written as , is the graph formed by taking G and H, and adding an edge between every vertex in and every vertex in H. This resulting graph is referred to the joint graph.

Figure 6 illustrates the graph obtained from the joint operation between whose vertices are colored red, and , with its single vertex colored blue.

A vertex v in graph G is referred to a center vertex of G if it is adjacent to each other vertex in the graph. In other words, a center vertex in G is one that is connected by an edge to all other vertices. The degree of , represents total number of edges that incident to v. If | is the total number of vertices in and w is a center vertex, then . A graph G that contains all of its center vertices is called a centered graph, denoted . A centered graph can be a basic graph or a graph produced from an operation. Some centered graphs for which we will determine their nonlocal-adjacency metric dimensions are: graphs and (including basic graphs), and graphs , and (including the graphs resulting from operations). From Figure 3, we can see that the only center vertex of graph is vertex . All vertices of graph are center vertices, and a center vertex of graphs , and is the vertex of , respectively.

The concept of adjacency distance is introduced by Jannesari and Omoomi as described in Definition 2.7.

Definition 2.7.[14]Given graph G, and let be an ordered set. For every vertex , the adjacency metric representation of v with respect to is with

where means v is adjacent to and means v is not adjacent to

The adjacency metric dimension of graph G is defined as follows.

Definition 2.8. [19]For an ordered subset and , the adjacency metric representation of with respect to is k-vector . The set is an adjacency metric resolving set with holds . The adjacency resolving set with the minimum number of vertices is called the adjacency basis of G. The adjacency metric dimension of written as dim , is defined as the number of elements in its adjacency basis.

Several researchers, as referenced in [19], have investigated and determined the adjacency metric dimensions for numerous types of connected graphs. For any graph , two distinct vertices p and q can either be adjacent or non-adjacent. Based on Definition 2.8, a resolving set in the context of nonlocal-adjacency metric only resolves between vertex pairs that are not adjacent. The minimum number of elements in such a resolving set is referred to the nonlocal-adjacency metric dimension, represented by . This study combines the adjacency metric resolving set concept and the nonlocal property in a graph. Thus, we can construct a concept that is produced from this development, as we can see in Definition 3.1.

In this section, we determine the nonlocal-adjacency metric dimension of basic graph , specifically non center vertex, that is path graph and cycle graph Theorem 3.2 describes nonlocal-adjacency metric dimension of

Theorem 3.2.Nonlocal-adjacency metric dimension of is

Proof. Let be labeled as shown in the Figure 1. The vertices set of is , and for .The proof is analyzed under two separate conditio

Case 1. For . We choose . The vertices in have adjacency metric representations with respect to

, but is adjacent to in . The cardinality of , for is minimum, because if we take a set with cardinality one, then but . In for , v3 , and So is not a nonlocal-adjacency metric resolving set of and is minimum, such that dim for Case 2. For . An ordered set is chosen with cardinality . The adjacency metric representation vertex with respect to is

for and . We have but and .Hence, all non

adjacent vertices in , have diferent adjacency metric representations. Therefore, is a nonlocal-adjacency metric resolving set of . The cardinality of is minimum. If we take any ordered subset with , and we choose , then there exists a vertex such that with

, and . Therefore, is a minimum nonlocal-adjacency metric resolving set of , the nonlocal-adjacency metric dimension of is dim .

Nonlocal-adjacency metric dimension of , is given in Theorem 3.3.

Theorem 3.3. Nonlocal-adjacency metric dimension of is

Proof. Suppose the graph is labeled as shown in Figure 2, so that the vertex set of is . An ordered subset is chosen, that is . It will be shown that is a nonlocal-adjacency metric resolving set of . The proof is divided into two cases, that is for odd and for even

Case 1 For odd

Choose the set . The ^{z⌊ n+1 ⌋

adjacency metric representation of with respect to is , with

for and . All vertices have distinct adjacency metric representations, except for and , which have the same representation , but is adjacent to . Hence, the adjacency

metric representations of non-adjacent vertices are all diferent. Therefore, is a nonlocal-adjacency metric resolving set for when and n is odd. The cardinality of is minimum because if an ordered subset

is taken with , say for some , then there exists a vertex such that

with , and

Thus, any ordered subset with is not a nonlocaladjacency metric resolving set of . Consequently, the nonlocal-adjacency metric dimension of is dim

Case 2 For , n even.

Choose the ordered set . The adjacency metric representation of the vertices with respect to is

for and

Hence, every vertex has a distinct nonlocal-adjacency metric representation. is a nonlocal-adjacency metric resolving set of The cardinality is minimum because if any ordered subset is taken with say , for some , then there exists a vertex such that with

and

Hence, any ordered subset with cannot serve as a nonlocal-adjacency metric resolving set for . Consequently, the nonlocaladjacency metric dimension of is given by dim , where n is even and □

Non local-adjacency metric dimension of centered graphs that are basic graphs are presented in Theorem 3.4 and Theorem 3.5. The following theorems, up to Theorem 3.8, address the nonlocal-adjacency metric dimension of centered graph that are the result of an operation. Characterization of nonlocal-adjacency metric dimension of a graph is presented in Theorem 3.4.

Theorem 3.4.Nonlocal-adjacency metric dimension of if and only if

Proof. (⇐) It is known that . From Definition 2.4, we know that all vertices in are adjacent to each other. there is no vertex in that is not adjacent to another. Hence, there is no vertex that can be chosen as a candidate for the element of the nonlocal-adjacency metric resolving set of Therefore, the nonlocal-adjacency metric resolving set of is the empty set. Thus, the minimum cardinality of the nonlocal-adjacency metric resolving set is zero or dim

(⇒) It is known that the nonlocal-adjacency metric dimension of is zero. This means the cardinality of the nonlocal-adjacency metric resolving set of is zero. Therefore, the nonlocal-adjacency metric resolving set of is the empty set. This means there is no vertex in that is not adjacent to another. Hence, all vertices in are adjacent to each other. The only connected graph that has this property is □

So, zero is lower bound of the nonlocal-adjacency metric dimension of graphs. This lower bound is sharp because is achieved by . Thus, we can write

Theorem 3.5 presents the nonlocal-adjacency metric dimension of star graph

Theorem 3.5.Nonlocal-adjacency metric dimension of is

Proof. Suppose the graph is labeled as shown in Figure 3. The vertex set of is , with deg for 2 and . Every vertex v with is not adjacent to one another. Let us choose the set . We will show that is a nonlocal-adjacency metric resolving set for . The adjacency metric representations of the vertices in with respect to the set are:

with

and for all , while for all

Hence, all pairs of non-adjacent vertices in have distinct representations. Therefore, is a nonlocal-adjacency metric resolving set for . The cardinality of is minimum because if we take any ordered subset with , and we choose then there is exist a vertex such that {zn−3

with . Therefore, is a minimum nonlocal-adjacency metric resolving set for . Hence, the nonlocal-adjacency metric dimension of is .

The next centered graph to be discussed is the fan graph. Let be a fan graph whose vertices are labeled as shown in Figure 6 .The vertex set of is , so the order of the graph is . The nonlocal-adjacency metric dimension of the graph is given in the following theorem.

Theorem 3.6.Nonlocal-adjacency metric dimension of fan graph 2 is

with

Proof. Let the vertices of graph labeled as , where is the vertex of degree and are the vertices of degree two, and are the vertices of degree three. Hence, the vertex set of is = . The proof is given in three cases.

Case 1.For m = 5.

Then , so

Choose the set . The vertices in have the following adjacency metric representations with respect to the set .

However, and . Therefore, T is a nonlocal-adjacency metric resolving set of , and the cardinality is minimum. Thus, the nonlocal-adjacency metric dimension of is dim

Case 2. For

Then , so

Choose the set . The vertices in have the following adjacency metric representations with respect to the set

Nevertheless, is adjacent to . Therefore, T constitutes a nonlocal-adjacency metric resolving set for the graph , and its cardinality is minimum. To justify minimumity, assume there exists a set with , for instance , where . In such a case, at least one pair of distinct vertices in the graph would share an identical nonlocal-adjacency metric representation with respect to

Thus, is not a nonlocal-adjacency metric resolving set. Therefore,

Case 3. For

Choose the ordered set . The vertices have the adjacency metric representations with respect to

with the distance defined by:

Hence, we obtain the adjacency metric representation with . While all other vertices in

have distinct representations. Thus, is a nonlocal-adjacency metric resolving set, and its cardinality is minimum, because if we choose any subset with will result the adjacency metric representasion with . Thus, is not a nonlocal-adjacency

metric resolving set. Therefore, we conclude dim .

Let be a wheel graph with vertices labeled as shown in the following figure.

The vertex set of is . The nonlocaladjacency metric dimension of is given in the following theorem.

Theorem 3.7.Nonlocal-adjacency metric dimension of , is

Proof. Proof is divided into two cases.

Case 1. For odd . Choose the ordered set . For every vertex in , the adjacency metric representation with respect to is

with

for and . Specifically for but v and are adjacent. Similarly for ) and , but the vertices with the same adjacency metric representation are still adjacent. For , but . It follows that the {z⌊ n+1 ⌋

adjacency metric representation of non-adjacent vertices in is all distinct. Consequently, is a nonlocal-adjacency metric resolving set of The cardinality of , is minimum, since if any ordered set with is taken, say then there exist vertices such that where , and , that is not being a nonlocal

adjacency metric resolving set of . Therefore, is the nonlocal-adjacency metric basis of , and , for odd .

Case 2. For even . Choose the ordered set The adjacency metric representation of the vertices in is

with

for and . For every , but . Therefore, the adjacency metric representation of non-adjacent vertices is distinct. Thus, T is a nonlocal-adjacency metric resolving set of . The cardinality of , is minimum, since if any ordered set with is taken, say , where , then there exist vertices such that , with , , and . Therefore, T is the nonlocal-adjacency metric basis of and , for even .

Given a complete graph with the vertex set and a graph with the vertex set . The graph is obtained by taking the graph and the graph . Then, every vertex in is connected to all the vertices in . The graph has vertex set and edge set . This can be expanded as .

As an illustration, the graph Km+Kncan be seen in Figure 8.

In Figure 8], the red vertex represents the vertex of and the green vertex represents the vertex of . The Nonlocal-adjacency metric dimension of with and , is presented below.

Theorem 3.8.Let be a complete graph with and let H be an empty graph . Nonlocal-adjacency metric dimension of is

Proof. Let . Adjacency metric representations of the vertices of with respect to are with and with

Since all of vertices that are all the same, which is equal to where are all adjacent to each other, then the adjacency metric representation of every two adjacent vertices is not diferent. However, for every with and Hence, although T serves as a nonlocal-adjacency metric resolving set for the graph , this does not establish it as a lower bound. Therefore, the nonlocal adjacency metric dimension satisfies dim.

Now, we show that dim be a nonlocal-adjacency metric resolving set with Assume that an ordered set is another minimum nonlocal-adjacency metric resolving set, or . If we select an ordered set so there is exist two vertices such that It should be noted that is

not a nonlocal-adjacency metric resolving set, which is contrary to the assumption. , the lower bound is dim . We conclude that .

This subsection explained definition of degree corona graph and the nonlocaladjacency metric dimension of the graph resulting from degree corona operation. Degree corona operation is an extension of the corona operation discovered by [4]. The following is the definition of a degree corona graph.

Definition 3.9.The degree corona graph of two graphs G and , denoted by , is a graph obtained by taking a graph G and copies of the graph H, denoted by (the copy of . For each vertex , connect to every vertex in each , where and

As an illustration, the degree corona graph can be seen in Figure Figure 9

In Figure 9, the red vertex represents the vertex of the graph and the blue vertex represents the vertex of the graph

Observation 3.10.Given a connected graph and H is a graph with at least two vertices. In any subgraph of , the vertices u and v are considered similar adjacency distance with respect to

Based on the above observation, we can formulate the lemma below.

Lemma 3.11.Let be a connected graph with order , and let H be a graph of order at least two such that H is not isomorphic to the complete graph . For each i, let ) represent the subgraph in associated with the copy of H.

Proof.

An observation regarding the element of basis of the degree corona graph was also obtained as follows.

Observation 3.12.Given a connected graph and H be an empty graph containing one or more vertices. In any subgraph H (the -copy of H) of the graph , the vertex is the element of basis of the graph

In addition to the definition of the degree corona graph, this research also presents a theorem on the nonlocal-adjacency metric dimension of the degree corona graph of centered graphs and the trivial graph

Theorem 3.13. Let be a centered graph, with and H is the trivial graph . The nonlocal-adjacency metric dimension of the degree corona graph

Proof. Based on Observation 3.12, let T be the nonlocal-adjacency metric resolving set of , that is but this is insuficient to establish it as the lower bound. So, the upper bound is . If the basis selection, aside from the vertex of , then it is not a basis.

Now, we show that dim . Let be a nonlocal-adjacency metric resolving set with . Assume that an ordered set is another minimum nonlocal-adjacency metric resolving set, or 1. If we select an ordered set 2 j, so that there exist two vertices such that . It should be noted that is not a

nonlocal-adjacency metric resolving set, which is contrary to the assumption. So, the lower bound is di . We conclude that .

Based on the above research results, the upper bound of the nonlocal-adjacency metric dimension is obtained as follows

In the modern era, where online shopping platforms are rapidly growing, the logistics distribution system has become a key focus for optimization by shipping and expedition companies. In a distribution network, when two major cities are directly connected through a main transportation route, the shipping schedule can be carried out routinely and eficiently. However, for cities that are not directly connected (non-adjacent), shipment planning must be done carefully; the logistics must pass through other cities, and the shipping schedules need to be adjusted to minimize travel time and costs. In determining the metric dimension of nonlocal-adjacency, the vertices in the graph represent the cities. Cities that have a direct transportation route are represented as adjacent vertices in the graph, while cities without a direct transportation route are represented as non-adjacent vertices. These non-adjacent vertices need to be resolved, as they require a transit warehouse for collecting logistics, which are then scheduled for shipment according to the delivery schedule. By modeling the distribution network as a graph and applying the concepts of adjacency and metric dimension, we can identify key points such as warehouse locations and transit warehouses that can improve overall distribution eficiency.