number.

Graph theory is a field of study in mathematics concerned with network structure and object interaction. Graph theory has numerous applications far beyond mathematics. Some of the significant applications of graph theory are in aviation networks, map directions, algorithms used by search engines, traffic signals, solving sudoku’s puzzles, epidemiology, and many more. Chemical graph theory is recognized as a wide rang field that combines mathematics and chemistry. It focuses on using mathematics to comprehend chemical structures by transforming them into graphs. Chemical structures are transformed into graphs by turning atoms into vertices and lines joining atoms into edges. Different parameters captivate the attention of many researchers to study chemical structures, and the identification of vertices is one of them.

The challenge of uniquely identifying the vertices in a graph attracted many researchers because of its widespread applications. Many researchers worked on uniquely identifying the vertices by using different useful concepts of graph theory including coloring, labeling, covering of vertices, and by definingιthe metric on graphs.

Vertex identification by defining metrics on graphs has numerous applications in chemical graph theory and computer networks including network discovery [5], navigation of robots [20], game strategies by using resolving sets in a hamming graph [12], to study digital images, resolving sets have been proposed in triangular, rectangular, and hexagonal grids [37] and coins weighing problems [46]. Afterwards, many researchers extended the idea of metric identification of vertices by determining different parameters of metric dimension including fractional [4], double [10], independent [11], weighted [14], k-metric [16], solid [17], local [40], mixed [19], connected [42] andιtheιstrongιmetricιdimension [45].

Consider aιconnectedιgraphιG with vertex set V (G). A path is a sequence of non-repeated vertices connected through edges. Distance is defined as the length of the shortest path connecting two vertices ψ, χιandιdenotedιbyιd(ψ, χ). Inspired by the issue of uniquely identifying the vertices of aιgraph,ιthe idea of locatingιsets was presented by Slater in [47]. After that Harary and Melter independently introduced a similar idea in [18] where they name locating sets as resolving sets. We use the term resolving sets throughout our work. A pair of vertices ξ, ζ ∈ V (G) resolves byιaιvertex x ∈ V (G)ιif distanceιbetween ξ and x isιnot equal to distance between ζ and x mathematically d(ξ, x) ̸= d(x, ζ). A set of vertices W in G said to be resolvingιset of Gιif each pairιof uniqueιvertices in G can be resolvedιby anyιvertex inιW. Cardinalityιof minimum resolvingιsetιisιcalled the metricιdimensionιofιG.

Theιstrongιmetricιdimensionιis more defined parameterιthanιtheιmetricιdimensionιwas originally introducedιin [45] by Sebö and Tannier and further studied by numerous researchers in many papers. That is aιvertexιx ∈ V (G)ιstrongly resolves twoιvertices a, b ∈ V (G)ιifιa shortestιpathιbetweenι(x, a) contains b or a shortest path between (x, b) contains a. A set ofιverticesιW ⊆ V (G) is calledιstrong resolvingιset of G if each pair of distinct vertices ofιG isιstrongly resolved byιany vertex in W the minimumιcardinalityιof overall strongιresolving sets of G isιcalledιthe strongιmetricιdimension ofιG,ιdenoted as Sdim(G) [45].

After SeböιandιTannier many researchers extend theιstudyιof theιstrongιmetricιdimension. The concept ofιstrongιmetricιdimensionιhasιbeen studied in some convex polytopes in [24] and in [43], in convex plane graph [1], in some famous graphs [27], in hamming graphs [25], in sunflower, friendship, helm and t-fold wheel graphs [26], in direct product graphs [28], in generalized sierpiriski graph [15], in power graph of finite groups [34], in anti-prism and king graphs [38], in zero divisor graph of rings [6], in distance hereditary graphs [35], in some graphs and their compliment [50], in cartesian sum graphs [31], in windmill graph Kn , sun graph Sn and mobius ladder graphs Mn [49], in generalized butterfly, starbarbell, and Cm ⊙ Pn graphs [36], in crossed prism [51].

Cartesianιproduct, directιproduct, rooted product,ιlexicographicιproduct, strong product and corona product are some common graph products, also called standard products. For such classes of graphs a lot of graphs invariant are studied to calculate their exact values or to predicting their action based on theιfactor graphs.ιThe strong metricιdimension of the graph’s products also has been investigated in the following published articles. In some families of direct product graph [28], rooted productιgraphs [30], lexicographicιproduct [29], strong product, corona product, graphs and digraphs, cartesian and direct product of graph has been investigated in [32, 33, 39, 41].

In this article, we investigate the strong metric dimension of the hollow coronoid structure. Different variants of metric dimension have been studied on hollow coronoid structures in the following articles. Metric and fault-tolerant metric dimension in [21], edge metric and fault-tolerant edge metric dimension in [22] and mixed metric dimension of hollow coronoid in [23]. Vertex identification through the definition of a metric on a graph has many applications in computer networks [52] and chemical graph theory [53, 54], such as network discovery, robot navigation, the study of digital images, resolving sets in triangular, rectangular, and hexagonal grids, and coin weighing problems. Subsequently, numerous scholars expand upon the concept of metric vertex identification by ascertaining various metric dimension characteristics, such as fractional, double, independent, weighted, k-metric, solid, local, mixed, linked, and strong metric dimensions. Additional uses include resource allocation in transportation networks [55, 56], researching the dissemination of information in social networks [57], identifying anomalies in cyber-physical systems, and examining epidemic trends in epidemiology.

In conclusion, this work creates a wealth of opportunities for additional theoretical research on the metric dimensions of chemical structures, algorithm development, and practical applications. Expanding upon these discoveries may reveal novel perspectives at the nexus of graph theory, metric geometry, and chemistry[58].

Now we are formalizing all the necessary terminology and notations that we will use throughout the article. LetιG denote a connectedιgraph shown in Figure 2, where the vertexιsetιofιG isιdenotedιbyιV (G)ιandιthe edgeιsetιofιG byιE(G). In a graph G if two vertices m, n share an common edge between them then they are called adjacent vertices and denoted by m ∼ n. A vertex set containing all the adjacent vertices of m is called neighborhood of m and denoted as N(m), sometimes N(m) is called open neighborhood of m. Aιvertex ϑ stronglyιresolvesιa pair (ϱ, φ) of distinct verticesιif either d(ϑ, ϱ) = d(ϑ, φ) +d(φ, ϱ) or d(ϑ, φ) = d(ϑ, ϱ) +d(ϱ, φ). In our work we take help from the following result of Kratica et al. [24] to find which pair of distinct vertices cannot be strongly resolves by any vertex of G.

Lemma 1. [24] Let (γ, δ) be a pairsιof verticesιin aιconnected graphιG such that d(γ, δ) ≥ d(ω, δ) ∀ ω ∈ N(γ) and d(γ, δ) ≥ d(ω, γ) ∀ ω ∈ N(δ) (1) Thenιthere isιno vertex inιV (G) − {γ, δ} which performs the strong metric identificationιfor the pair (γ, δ).

To examine theιstrongιmetricιdimensionιof hollow coronoid weιuse the concept ofιstrong resolvingιgraph which was proposed by Ollermann andιFransen in [39]. If a pair (ξ, ζ) ∈ V (G) of distinct vertices satis-fied both the condition in (1) then the vertices areιsaidιtoιbe mutuallyιmaximallyιdistant,ιdenoted as ξM M Dζ. Theιstrongιresolvingιgraphιof a graphιG is aιgraph GSR withιvertexιset contained all M M D vertices and there is an edge between two vertices if and if they are M M D with each others in original graph G.

If W ⊆ V (G) then W is aιvertexιcoverιofιGιif everyιedgeιinιG isιincidentιwith someιvertex of W and the cardnality ofιminimum vertexιcover of G is called vertex cover number or covering number of G denotedιbyιτ(G) [48]. In the following theorem Ollerman and Fransen established a relationship between covering number of strong resolving graph GSR and the strongιmetric dimension.

Theorem 2. [39] For anyιconnected graph G, sdim(G) = τ(GSR).

Throughout our work notation Kl represent complete graph and Cl represent cycle graph. We have the following remark according to the definition of vertex cover number.

Remark 3. Assume that G is a connected graph, then

(1) If G is complete graph Kl on l ≥ 2 vertices, then τ(G) = l − 1.

(2) If G is cycle graph on l ≥ 3 vertices, then τ(G) = ⌈ ⌉.

The name coronoid was coined in 1987 [9] due to its potential link with benzenoid. Since coronoid is a benzenoid with a central hole. A particular kind of polyhex system having origins in organic chemistry is referred to

′ ′ ′

as a "coronoid". Hollow coronoid contained six sides x, y, z, x , y , z as shown in Figure 3. It drives from the catacondensed coronoids and is a member of the primitive coronoid [7].

A new type of hollow coronoid structure has been identified in [44]. In [8, 2, 3] author discussed mathematical study of coronoid and related structures. [13] describes the relationship between the hollow coronoid and the polyhex..

′ ′ ′

Our study examines a six-sided x = x = y = y = z = z hollow coronoid, however the three sides (x, y, z) are

′ ′ ′

symmetric to the other three sides (x , y , z ). Hollow coronoid label as HC(x, y, z) with x, y, z ≥ 2. Hollow coronoid contained total 8(x + y + z − 3) vertices and 10(x + y + z − 3) edges from which 4(x + y + z − 3) are of degreeι2 andιthe same number ofιvertices with degreeι3. The description of the vertexιand edgeιset of a hollowιcoronoidιstructureιHC(x, y, z) is given as:

V (HC(x, y, z)) = {pi, pi : 1 ≤ i ≤ 2x − 1} ∪ {si, si : 1 ≤ i ≤ 2z − 1} ∪ {qi, qi : 1 ≤ i ≤ 2y − 1} ∪ {ri, ri : 1 ≤ i ≤ 2x − 3} ∪ {ti, ti : 1 ≤ i ≤ 2z − 3} ∪ {ui, ui : 1 ≤ i ≤ 2y − 3} ,

′ ′ ′ ′

E(HC(x, y, z)) = pipi+1, pipi+1 : 1 ≤ i ≤ 2x − 2 ∪ qiqi+1, qiqi+1 : 1 ≤ i ≤ 2y − 2

′ ′ ′ ′

∪ sisi+1, sisi+1 : 1 ≤ i ≤ 2z − 2 ∪ riri+1, riri+1 : 1 ≤ i ≤ 2x − 4

′ ′ ′ ′

∪ uiui+1, uiui+1 : 1 ≤ i ≤ 2y − 4 ∪ titi+1, titi+1 : 1 ≤ i ≤ 2z − 4

′ ′ ′ ′

∪ tisi+1, tisi+1 : 1 ≤ i(odd) ≤ 2z − 3 ∪ ripi+1, ripi+1 : 1 ≤ i(odd) ≤ 2x − 3

′ ′ ′ ′ ′ ′

∪ uiqi+1, uiqi+1 : 1 ≤ i(odd) ≤ 2y − 3 ∪ q1s2z−1, qis2z−1, s1p1, s2z−1p2x−1

′ ′ ′ ′ ′ ′ ′ ′

∪ p1q1, p2x−1q2y−1, t2z−3u1, t1u2y−3, t1r1, t2z−3r2x−3, u1r1, r2x−3u2y−3 .

Lemma 4. The strong resolving graph of HC(x, y, z) where x = y = z ≥ 2 is isomorphic to (x+y+z−6)K4+3C4. Proof : In order to construct HC(x, y, z)SR we find each pair (u, v) of vertices in HC(x, y, z) such that uM M Dv. For the M M D pairs of vertices consider the table 1 It follows according to Table 1 all pairs of vertices satisfies

(pi, pj), i ̸= j for i, j ∈ {1, 2x − 1} d(pi, pj) = 4x − 1 d(pi, a) = 4x − 2 d(b, pj) = 4x − 2

′ ′ ′

(si, qj), i ̸= j for i, j ∈ {1, 2z − 1} d(si, qj) = 4z − 1 d(si, a) = 4z − 2 d(b, qj) = 4z − 2

′ ′ ′

(qi, sj), i ̸= j for i, j ∈ {1, 2y − 1} d(qi, sj) = 4y − 1 d(qi, a) = 4y − 2 d(b, sj) = 4y − 2

both the condition in (1) so, vertices in each pairs are M M D with each other.

Now, consider the pair (pi, pi) for 1 ≤ i(odd) ≤ 2x − 1. Note that, ′ 6x − 7 , 3 ≤ i(odd) ≤ 2x − 3, d(pi, pi) = 6x − 5 , i = {1, 2x − 1}.

and for a ∈ N(pi), b ∈ N(pi) ′ 6x − 8 , 3 ≤ i(odd) ≤ 2x − 3, d(pi, a) = d(b, pi) = 6x − 6 , i = {1, 2x − 1}.

′ ′ ′ ′

the pair (pi, pi) satisfied both the conditions in (1) as d(pi, a) = d(b, pi) ≤ d(pi, pi). So, piM M Dpi with each other.

Now, consider the pair (qi, si) for 1 ≤ i(odd) ≤ 2y − 1. Note that, ′ 6y − 7 , 3 ≤ i(odd) ≤ 2y − 3, d(qi, si) = 6y − 5 , i = {1, 2y − 1}.

and for a ∈ N(si), b ∈ N(qi) ′ 6y − 8 , 3 ≤ i(odd) ≤ 2y − 3, d(qi, a) = d(b, si) = 6y − 6 , i = {1, 2y − 1}.

′ ′ ′ ′

the pair (qi, si) satisfied both the conditions in (1) as d(qi, a) = d(b, si) ≤ d(qi, si). So, qiM M Dsi with each other.

Now, consider the pair (si, qi) for 1 ≤ i(odd) ≤ 2z − 1. Note that, ′ 6z − 7 , 3 ≤ i(odd) ≤ 2z − 3, d(si, qi) = 6z − 5 , i = {1, 2z − 1}.

and for a ∈ N(qi), b ∈ N(si) ′ 6z − 8 , 3 ≤ i(odd) ≤ 2z − 3, d(si, a) = d(b, qi) = 6z − 6 , i = {1, 2z − 1}.

′ ′ ′ ′

the pair (si, qi) satisfied both the conditions in (1) as d(si, a) = d(b, qi) ≤ d(si, qi). So, siM M Dqi with each other. We construct the following vertex and edge set of HC(x, y, z)SR from M M D pairs of vertices in HC(x, y, z).

′ ′ ′ ′ ′ ′

V (HC(x, y, z)SR) = {p1, p3, . . . , p2x−1, p1, p3, . . . , p2x−3} ∪ {s1, s3, . . . , s2z−1, s1, s3, . . . , s2z−3}

′ ′ ′ ′ ′ ′

∪ {q1, q3, . . . , q2y−1, q1, q3, . . . , q2y−3} ∪ {r2, r4, . . . , r2x−4, r2, r4, . . . , r2x−4}

′ ′ ′ ′ ′ ′

∪ {t2, t4, . . . , t2z−4, t2, t4, . . . , t2z−4} ∪ {u2, u4, . . . , u2y−4, u2, u4, . . . , u2y−4}

′ ′ ′ ′ ′ ′

E(HC(x, y, z)SR) = {p1 ∼ p1, p3 ∼ p3, . . . , p2x−1 ∼ p2x−1} ∪ {s1 ∼ q1, s3 ∼ q3, . . . , s2z−1 ∼ q2y−1}

′ ′ ′ ′ ′ ′

∪ {q1 ∼ s1, q3 ∼ s3, . . . , q2y−1 ∼ s2z−1} ∪ {p3 ∼ r2, p5 ∼ r4, . . . , p2x−3 ∼ r2x−4}

′ ′ ′ ′ ′ ′

∪ {s3 ∼ u2, s5 ∼ u4, . . . , s2z−3 ∼ u2y−4} ∪ {q3 ∼ t2, q5 ∼ t4, . . . , q2y−3 ∼ t2z−4}

′ ′ ′ ′ ′ ′

∪ {r2 ∼ r2, r4 ∼ r4, . . . , r2x−4 ∼ r2x−4} ∪ {t2 ∼ u2, t4 ∼ u4, . . . , t2z−4 ∼ u2y−4}

′ ′ ′ ′ ′ ′

∪ {u2 ∼ t2, u4 ∼ t4, . . . , u2y−4 ∼ t2z−4} ∪ {r2 ∼ p3, r4 ∼ p5, . . . , r2x−4 ∼ p2x−3}

′ ′ ′ ′ ′ ′

∪ {t2 ∼ q3, t4 ∼ q5, . . . , t2z−4 ∼ q2y−3} ∪ {u2 ∼ s3, u4 ∼ s5, . . . , u2y−4 ∼ s2z−3}

′ ′ ′ ′ ′ ′

∪ {p1 ∼ p2x−1, p2x−1 ∼ p1, s1 ∼ q2y−1, s2z−1 ∼ q1, q1 ∼ s2z−1, q2y−1 ∼ s1}.

Theorem 5. For x = y = z ≥ 2, sdim(HC(x, y, z)) = 3(x + y + z) − 12.

Proof : As strong resolving graph of HC(x, y, z) is isomorphic to (x+y+z−6)K4 +3C4, by Remark 3 a minimum vertex cover W of the graph HC(x, y, z)SR must contains 3 vertices from each copies of K4 and 2 vertices from each copies of C4. So, by Theorem 2 sdim(HC(x, y, z)) = τ(HC(x, y, z)) = |W| = 3(x + y + z) − 12. □

The strong metric dimension is a useful notion with various applications in many fields, including networks security, robotics, wireless sensor networks, GPS navigation system, fault tolerance and network disruption. In this article, weιextendedιtheιstudyιofιthe strong metricιdimension by finding the strong metric dimension of HC(x, y, z) for x = y = z ≥ 2 by establishing a relationship between covering number of strong resolving graph HC(x, y, z)SR and the strong metric dimension. We concludeιthat sdim(HC(x, y, z)) = 3(x + y + z) − 12. In future we extend our work by findingιthe strongιmetric dimension of hollow coronoid when x, y, z are not equal to each other and further by studying other invariant of metric dimension.

This paper presents an important advancement in studying the strong metric dimension of hollow coronoid structures. However, there are several promising directions for future work:

(1) Investigate the strong metric dimension of hollow coronoid structures where x, y, and z are not equal.

This paper focused on the case where the three sides of the hollow coronoid structure are symmetric. Analyzing asymmetric hollow coronoids where x ̸= y ̸= z would be an interesting extension.

(2) Explore other metric dimension invariants of hollow coronoids. In addition to the strong metric dimension, parameters like fractional metric dimension, mixed metric dimension, k-metric dimension etc. could provide further insights into these chemical structures when transformed into graphs.

(3) Examine other classes of chemical structures. The techniques used in this paper could be applied to find the strong metric dimension of other chemicals like benzenoids, phenylenes, polyhexes etc. This would expand our understanding of chemical graphs.

(4) Consider weighted graphs. Assigning weights to edges of the graph representation could reveal new findings regarding resolving sets and metric dimensions. The interplay between weights and shortest paths may impact the strong metric dimension.

(5) Develop computational methods and algorithms. Efficient methods to calculate the strong metric dimension for large chemical graphs would enable rapid analysis. Approximation algorithms and heuristics could produce useful results at scale.

(6) Explore applications of the strong metric dimension. This parameter has diverse applications in areas like robot navigation, network discovery, fault diagnosis etc. Relating the results to practical uses and implementations would demonstrate the utility of this research.

In summary, this work opens up many possibilities for further theoretical studies of metric dimensions of chemical structures, development of algorithms, and real-world applications. Building on these findings could uncover new insights at the intersection of graph theory, metric geometry and chemistry.

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