Studies of Chordal Ring Networks via Double Metric Dimensions

Locating the sources of information spreading in networks including tracking down the origins of epidemics, rumors in social networks, and online computer viruses, has a wide range of applications. In many situations, identifying where an epidemic started, or which node in a network served as the source, is crucial. But it can be dicult to determine the root of an outbreak, especially when data are scarce and noisy. e goal is to nd the source of the infection by analysing data provided by only a limited number of observers, such as nodes that can indicate when and where they are infected. Our goal is to investigate where the least number of observers should be placed, which is similar to how to gure out the minimal doubly resolving sets in the network. In this paper, we calculate the double metric dimension of chordal ring networks CRn(1, 3, 5) by describing their minimal doubly resolving sets.


Introduction and Preliminaries
Suppose that Γ is a connected, nite, and undirected graph. Its vertex set is V Γ and its edge set is E Γ . e distance d (f, g) between two vertices f, g ∈ V Γ in a connected graph is the shortest path connecting those two vertices. If e ∈ V Γ is a vertex such as d(e, f) ≠ d(e, g), then we say e resolves the vertices f and g. e representation r(h, J Γ ) of a vertex h with respect to an ordered set J Γ j υ |1 ≤ υ ≤ ρ ⊆V Γ is de ned as a ρ-vector (d(h, j 1 ), . . . , d(h, j ρ )), also called the distance vector. If each vertex of the graph Γ has a unique distance vector with respect to the set J Γ , then J Γ is called the resolving set (or sometimes called locating set) for graph Γ. e minimum possible cardinality of a resolving set in Γ is called the metric dimension (MD), represented as dim(Γ). A resolving set of cardinality dim(Γ) is said to be the metric basis of Γ. e notion of MD was rst proposed by Slater in [1] and investigated independently by Harary and Melter in [2] because of the problem of locating an intruder in a network. Later, Chartrend and Zang [3] demonstrated various applications in biology, robotics, and chemistry. From a twodimensional real plane, the MD of graphs can extend the concept of trilateration. For instance, distances are used by the Global Positioning System (GPS) to identify an object's location on Earth. In the context of MD, Hamming graphs are closely related to the di culties in weighing of coins discussed in [4,5], and the comprehensive study of the Mastermind game given in [6].
Resolvability of graphs has become a signi cant parameter in graph theory as a result of its widespread applicabilities in di erent areas of mathematics, including facility location problems, network discovery and veri cation [7], applications in molecular chemistry [8], the positioning of robots in a network [9], routing protocols geographically [10], the problems of sonar or LORAN stations [1], and the optimization problem in combinatorics [11]. After that, we will have a look at some research work on the mathematical signi cance of this distance-based parameter. e MD has been used to study a variety of mathematically interesting graph families. We will highlight some of the signi cant work in this section: Similar to many other graph-theoretic parameters, nding the MD of arbitrary graphs is a computationally tough task [9,12]. For example, the bounds for the MD of the Petersen graph family was studied by Shao et al. [13]. On various distance-regular graphs (such as kayak paddle graphs and chorded cycles), Ahmad et al. investigated the MD [14]. Bailey et al. [15] investigated the MD of Kneser graphs. For wheel graphs, Buczkowski et al. in [16] investigated the MD, whereas Baca et al. in [17] examined the MD of regular bipartite graphs. Chartrand et al. [8] categorized n-vertex graphs with MD 1, n − 2 and n − 1. From the perspective of MD, graphs of relevance in group theory, such as Cayley digraphs [18] and Cayley graphs formed by certain finite groups [19], have also been investigated. [20] provides a response to the question of whether the MD is a finite number or an infinite quantity. e minimum ordered resolving sets of Jahangir graphs (resp. necklace graphs) were investigated by Tomescu et al. in [21] (resp. in [22]). Investigations have also been carried out into the MD and the resolving sets of product graphs, such as the categorical product of graphs [23] and the cartesian products [24].
MD has also been generalised and extended by offering more mathematically rigorous general ideas, such as the double metric dimension (DMD). Caceres et al. [24] proposed and defined the notion of doubly resolving sets (DRSs) of graph Γ as follows: a pair of vertices g and h of graph Γ is said to double resolve vertices g′ and h ′ if the following equation holds: where every two different vertices of Γ are resolved by some two vertices of N Γ . e minimal doubly resolving set (MDRS) problem is to find a DRS of Γ with the smallest cardinality, which is called the DMD of Γ indicated by ψ(Γ). If the vertices g ′ and h ′ can be doubly resolved by the vertices g and h then either Because a DRS is also a resolving set, we have ψ(Γ) ≥ dim(Γ), so we can use the MDRSs to get an upper bound on the MD of the graph under discussion. e idea of establishing upper boundaries in the cartesian product encouraged us to explore on DRSs of different graph classes. In general graphs, the MDRS problem has been shown to be NPhard [25]. Many families of graphs, such as cocktail graphs, prisms, and jellyfish graphs, have been studied for the problem of finding the MDRSs (for details see: [26,27]). e MDRSs for convex polytope structures and Hamming graphs have been derived and may be found in [28], and [29], respectively. e DMD and minimal order resolving sets of Harary and circulant graphs were studied in [30,31]. It was Chen et al. that provided the first approximated upper bounds for the MDRS problem [32]. e authors in [33,34] demonstrated that the DMD of some convex polytope structures is finite and constant. e line graphs of chorded cycles [35], kayak paddle graphs [36], n-Sunlet, and prism graphs [37] were discovered to have the MD and DRSs. In [38], layersun graphs and associated line graphs were studied for the MDRSs. Liu et al. gave the results for the minimum order resolving sets and DMD of the line graph of the Necklace graph in [39].
Using resolving sets is a natural way to find the origins of a network spread. For example, determining where a disease originated as it spreads through a population might be helpful in a variety of scenarios. A direct solution can be found if the time at which the spread began and the internode distances are reliable and known. Resolvability, however, needs to be expanded to account for things like an unknown start time and irregular transmission delays among nodes. e former problem can be solved by employing DRSs. e variation involved with arbitrary internode distances is critical in successfully determining the source of a spread [40].
Detecting virus sources in starlike networks are more complicated than in pathlike networks. While the DMD is n − 1 for a star-type structure with n nodes, and for a pathtype structure with the same number of nodes it is 2 (see [40]). In addition, this demonstrates that the DMD is always reliant on the topology of the network being used. e purpose of this research is to find the MDRSs and DMD for a class of chordal ring networks, which are helpful in the designing of local area networks. e following result regarding the MD of chordal ring networks CR n (1, 3, 5) is helpful in determining its DMD:  (1,3,5) In this section, we calculated the MDRSs and DMD for chordal ring network CR n (1,3,5). If every closed path of length 4 or more in an undirected network contains a chord, it is referred to as chordal. Arden and Lee [42] first proposed chordal ring networks of degree 3. An undirected cycle of even order can be used to form a chordal ring network, in which all newly added chords connect an even labelled node to an oddly labelled node.
Let α, β, c ∈ 1, 2, . . . , n − 1 { } be three distinct odd integers and n ≥ 3 be an even natural number. A chordal ring network having order n with chords α, β, and c represented by CR n (α, β, c), has the nodes set Z n , and links are given by δ ∼ δ + α, δ ∼ δ + β and δ ∼ δ + c for every even node δ ∈ Z n . By definition, every chordal ring network CR n (α, β, c) is bipartite and 3-regular. Further, every odd node θ ∈ Z n forms a link with θ − α, θ − β, θ − c in CR n (α, β, c). ere are no specific rules for selecting three different odd integers α, β, and c from the set 1, 2, . . . , n − 1 { }. Actually, we can choose any three different odd integers and, as a result, we obtain different chordal ring networks each time, with the possibility that few of them are isomorphic. To make a CR n (α, β, c), select 3 different even integers from the set 1, 2, . . . , n − 1 { } and connect odd numbered nodes to them, as explained earlier. In addition, α, β, and c are stated to be chords because they help to connect nodes in the CR n (α, β, c), and each obtained link represents a chord for at least one path in CR n (α, β, c) as demonstrated in Figure 1.
De ne S υ (h 0 ) h ∈ V CR n (1,3,5) : d(h 0 , h) υ be a vertex set in V CR n (1,3,5) at distance υ from h 0 . Table 1 is simply constructed for S υ (h 0 ) and employed to compute the distance between any pair of vertices in V CR n (1,3,5) . e symmetry of CR n (1, 3, 5) for any even integer n ≥ 6, shows the following fact that: As a conclusion, by knowing the distance d(h 0 , h) for all h ∈ V CR n (1,3,5) , we can reconstruct the distances between every two vertices in V CR n (1,3,5) .

Conclusion
is study is concerned with the concept of calculating MDRSs of graphs that has been proposed earlier in the literature. e DMD of chordal ring networks CR n (1,3,5) is computed by describing their MDRSs. In this study, we found that the number of vertices in the chordal ring networks does not affect its DMD.

Data Availability
e whole data are included within this article. However, the reader may contact the corresponding author for more details on the data.

Conflicts of Interest
e authors declare that there are no conflicts of interest.