Bounds on the partition dimension of one pentagonal carbon nanocone structure

1 Introduction

The term metric dimension or in some literature employed as the metric index is widely studied in recent times due to the applicability in numerous fields, including chemistry of pharmaceutics (Nadeem et al., 2020; Saenpholphat and Zhang, 2003), robot’s navigation (Iijima and Ichihashi, 1993). Moreover, solar coast-Loran stations (Brinkmann and Cleemput, 2011), and in optimization employing combinatorics techniques (Johnson, 1993). When this concept is implemented in pharmaceuticals problems in drug design, mathematical representations for a set of chemical compounds are needed to find distinct representations for distinct compounds. A chemical compound structure can be modified into a labeled chemical network whose nodes and links are atoms and atom-bonds, respectively. The minimum number of nodes (atoms) provides the required distinct representations to distinct nodes (atoms) in the network due to the metric dimension concept (Chartrand et al., 2000).

Let $G=(V,E)$ be the connected graph. The distance between two vertices $i,j$ of graph $G$ is the length of the shortest path between them, and is usually denoted by $d(i,j)$ . Let $S\subset V\left(G\right)$ the distance between $i\in V\left(G\right)$ and $S$ is defined as $d(i,S)=min\left\{d\right(i,k):k\in S\}$ . Let $R=\{r_1,r_2,\cdots ,r_n\}$ be an ordered set of $V\left(G\right)$ . The metric codes of a vertex $i\in V\left(G\right)$ with respect to $R$ is written as $r\left(i\right|R)=(d(i,r_1),d(i,r_2),\cdots ,d(i,r_n)\}$ . The set $R$ is usually known as resolving (distinguishing, determining) set of $G$ if the metric codes of any two vertices of $G$ are unique corresponding to $R$ . The least number of elements within the distinguishing set of a graph $G$ is called metric dimension, the usual notation in the literature is $\mathit{dim}\left(G\right)$ , (Slater, 1975).

Chemical graph theory aids interpretation of a chemical graph's nuclear assistant properties. Chemical graph theory is expected to play an important role in showcasing and organizing any constructed framework or component organize out. A chemical compound or graph is a graphical portrayal that includes atoms called vertices and atom-to-atom bonds known as edges.

Metric basis are very useful in the reshaping of a chemical network, therefore in the literature there is intensive research are available on different chemical structures, hollow coronoid structure is studied in this term (Mehreen et al., 2018), the necklace graph with these properties are discussed in (Khuller et al., 1996). The resolving set for the graph obtained from categorical product of two graphs available in (Arockiaraj et al., 2018), where a general circulant graphs are discussed in (Sebö and Tannier, 2004). Some of the results concerning metric dimension are detailed in (Estrada-Moreno, 2020; Liu et al., 2020; Valentini and Ciambella, 2016).

The concept of metric and partition dimension are closely related, as partition dimension is the generalization of the metric dimension. Instead of computing the distance between two vertices, we do it between a vertex and a set. The concept of partition dimension was introduced by (Chartrand et al., 2000).

Let $\sqcap =\{{\sqcap }_1,{\sqcap }_2,\cdots ,{\sqcap }_{\mathcal{l}},\}$ be the partition of a connected graph. Now for a vertex $j\in V\left(G\right)$ the partition dimension with respect to $\sqcap$ is $r\left(j\right|\sqcap )=(d(j,{\sqcap }_1),d(j,{\sqcap }_2),\cdots ,d(j,{\sqcap }_{\mathcal{l}}))$ , where $d(j,{\sqcap }_w)=min\{d(j,y):y\in {\sqcap }_w\}$ for $1\le w\le \mathcal{l}$ . The distinct codes of the two vertices $i,j\in V\left(G\right)$ with respect to $\sqcap$ , that is $r\left(i\right|\sqcap )\ne r(j|\sqcap )$ , such a partition $\sqcap$ is known as distinguishing partition of $G$ (Koam and Ahmad, 2020; Alipour and Ashrafi, 2009).

Several authors have published articles related to the partition dimension of graphs; book graph are discussed in (Iijima, 1991) in terms of this concept, the resulted graph from the operation of comb product are detailed in (Ahmad and Sultan, 2017), generalized version of partition dimension are studied in (Imran et al., 2018; Ge and Sattler, 1994), and a generalized class of graphs are studied in terms of partition dimension in (Nadeem et al., 2021). The partition dimension for the families of circulant graphs, multipartite, and chemical graphs of fullerene are discussed in (Hussain et al., 2019; Baskoro and Haryeni, 2020). The graphs of convex polytopes for their partition dimension are in (Wei et al., 2021, 2021,; Nadeem et al., 2021; Hussain et al., 2020). The graph with partition dimension $\left|V\right|-3$ discussed (Akhter and Farooq, 2019).

The partition dimension, usually denoted as $\mathit{pd}\left(G\right)$ , is the least number of elements residing in the distinguishing set $\sqcap$ of $G$ . In general for a connected graph $G$ , we have. $$\mathit{pd}\left(G\right)\le dim\left(G\right)+1.$$

The emphasis of this study is finding the partition dimension problem of the $1$ -pentagonal carbon nanocone structure. In combinatorial chemistry, the set of atoms, molecules, and compounds are represented mathematically in a unique manner for a large structure. The vertices and edges of a graph depict the atoms and bond types, respectively.

As mentioned, the partition dimension is the generalization of the metric dimension, so it would be interesting to find the partition dimension. This article was motivated by the results for the metric dimension of $1$ -pentagonal carbon nanocanes; this article found the partition dimension’s sharps bounds.

2 1-pentagonal carbon nanocone structure $\mathit{C}\mathit{N}{\mathit{C}}_{\mathit{\zeta },5}$

Carbon nanocone (CNC) started its journey in $1994$ (Amrullah et al., 2015), when it was treated in experimentation as $19^{°}$ free standing cones, and it is found in the tunneling microscopy. It gained much attraction in the current era and came out in different topologies, such as carbon nanohorns (Vetrik and Ahmad, 2017; Bultheel and Ori, 2018). There is another construction of nanocone by putting a one-end cap on the nanotube (Cataldo et al., 2015; lijima et al., 1992; Justus, 2007). Viewing oppositely, a nanocone with six pentagons contained an opening with $0^{°}$ angle, which looks like a nanotube with one side of the face closed (capped). Moving towards one pentagonal nanocone observed or came into existence in the same year of $1994$ (Amrullah et al., 2015). It is a proven statement that this particular type of nanocone is constructed from graphene structures. Removal of graphene is a $60^{°}$ wedge, and after the joining of its edges and formed a shape of a cone with a pentagonal defect on the side of the apex. For different study related to the carbon nanotubes, graphene sheets, and nanocones are available in (Azizkhani et al., 2020; Kolahchi and Kolahdouzan, 2021; Al-Furjan et al., 2020; Al-Furjan et al., 2021; Al-Furjan et al., 2021; Kolahchi et al., 2021; Faramoushjan et al., 2021).

It is widespread with its applicative point of view after discovering different types of nanocones and their topologies. The possible applications in energy storage, gas storage, gas sensors, biosensors, and chemical probes (Koam et al., 2021; Darmaji and Alfarisi, 2017; Maritz and Vetrik, 2017). Nanocones are carbon networks that can be represented as infinite cubic planar graphs. Authors in (Ahmad et al., 2018) investigated the topological modeling techniques of one pentagon carbon nanocones. Topological properties of carbon nanocones in (Ahmad et al., 2020), and computation calculation of wiener index in (Chu et al., 2020).

The Fig. 1 shows the molecular structure of nanocone, and it can be represented graphically by showing into vertices and edges form, which is in Fig. 2. As shown in Fig. 2 that the centermost cycle of with $5$ vertices named as the first cycle, and the vertex set is defined as $V\left(CN{C}_{\zeta ,5}\right)=\{{\psi }_{1,j},:1\le j\le 5\}\cup \{{\psi }_{2,j},:1\le j\le 15\}\cup \{{\psi }_{3,j},:1\le j\le 25\}\cup \{{\psi }_{4,j},:1\le j\le 35\}\cup \{{\psi }_{i,j},:1\le ∊\le \zeta ,1\le j\le 10∊-5\}\cup \{{\psi }_{\zeta ,j},:1\le j\le 10∊-5\}.$

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Fig. 1

Pentagonal Nanocone Structure.

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Fig. 2

One pentagonal nanocone: The labeling stated is the subscript or index of vertices. For example the $1,1$ is represented as ${\psi }_{1,1}$ in the results.

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The graph of $\mathit{CN}{C}_{\zeta }\left[n\right]$ comprises the conical structure possessing a cycle of length $n$ and $\zeta$ layers of hexagons lying on the conical surface around its center. This article focuses primarily on the case when $n=5$ . The number of hexagons is represented as $\zeta$ incorporating the conical surface, and the number $5$ indicates a pentagon lies on the tip known as the core. The metric dimension for $\mathit{CN}{C}_{\zeta ,5}$ is obtained as;

Now the following theorem is the sharp upper bounds on the partition dimension of one pentagonal nanocone structure $\mathit{CN}{C}_{\zeta ,5}.$ $\mathit{CN}{C}_{\zeta ,5}.$

Proof. To show that the partition dimension of $\mathit{CN}{C}_{\zeta ,5}$ is $\le 4,$ suppose the resolving partition set $\sqcap =\{{\sqcap }_1,{\sqcap }_2,{\sqcap }_3,{\sqcap }_4\}$ where ${\sqcap }_1=\left\{{\psi }_{\zeta ,1}\right\},$ ${\sqcap }_2=\left\{{\psi }_{\zeta ,3}\right\},$ ${\sqcap }_3=\left\{{\psi }_{\zeta ,2\zeta +1}\right\},$ ${\sqcap }_4=V\left(CN{C}_{\zeta ,5}\right)\backslash \{{\psi }_{\zeta ,1},{\psi }_{\zeta ,3},{\psi }_{\zeta ,2\zeta +1}\}.$ The representations for the vertices of $\mathit{CN}{C}_{\zeta ,5}$ are:

When $1\le j\le 5$ , the codes for the vertices with respect to the partition resolving set are following, in which the very first is the representations of the vertices of first and second cycle;

$r\left({\psi }_{1,1}|\sqcap \right)=\left(2\zeta -3,2\zeta -1,2\zeta -1,0\right),$ $r\left({\psi }_{1,2}|\sqcap \right)=\left(2\zeta -2,2\zeta -2,2\zeta -2,0\right),$

$r\left({\psi }_{1,3}|\sqcap \right)=\left(2\zeta -1,2\zeta -1,2\zeta -1,0\right),r\left({\psi }_{1,4}|\sqcap \right)=\left(2\zeta -1,2\zeta ,2\zeta ,0\right),$

$r\left({\psi }_{1,5}|\sqcap \right)=\left(2\zeta -1,2\zeta ,2\zeta ,0\right).$

$r\left({\psi }_{2,1}|\sqcap \right)=\left(2\zeta -4,2\zeta -2,2\zeta ,0\right),$ $r\left({\psi }_{2,2}|\sqcap \right)=\left(2\zeta -5,2\zeta -3,2\zeta -1,0\right),$

$r\left({\psi }_{2,3}|\sqcap \right)=\left(2\zeta -4,2\zeta -4,2\zeta -2,0\right),$ $r\left({\psi }_{2,4}|\sqcap \right)=\left(2\zeta -3,2\zeta -3,2\zeta -3,0\right),$

$r\left({\psi }_{2,5}|\sqcap \right)=\left(2\zeta -2,2\zeta -2,2\zeta -4,0\right),$ $r\left({\psi }_{2,6}|\sqcap \right)=\left(2\zeta -1,2\zeta -1,2\zeta -3,0\right),$

$r\left({\psi }_{2,7}|\sqcap \right)=\left(2\zeta ,2\zeta ,2\zeta -2,0\right),$ $r\left({\psi }_{2,8}|\sqcap \right)=\left(2\zeta +1,2\zeta +1,2\zeta -1,0\right),$

$r\left({\psi }_{2,9}|\sqcap \right)=\left(2\zeta +1,2\zeta +2,2\zeta ,0\right),$ $r\left({\psi }_{2,10}|\sqcap \right)=\left(2\zeta ,2\zeta +1,2\zeta +1,0\right),$

$r\left({\psi }_{2,11}|\sqcap \right)=\left(2\zeta +1,2\zeta +2,2\zeta +2,0\right),$ $r\left({\psi }_{2,12}|\sqcap \right)=\left(2\zeta ,2\zeta +2,2\zeta +2,0\right),$

$r\left({\psi }_{2,13}|\sqcap \right)=\left(2\zeta -1,2\zeta +1,2\zeta +1,0\right),$ $r\left({\psi }_{2,14}|\sqcap \right)=\left(2\zeta -2,2\zeta ,2\zeta +2,0\right),$

$r\left({\psi }_{2,15}|\sqcap \right)=\left(2\zeta -3,2\zeta -1,2\zeta +1,0\right).$

Representations of the vertices of third cycle;

$r\left({\psi }_{3,1}|\sqcap \right)=\left(2\zeta -6,2\zeta -4,2\zeta -1,0\right),$ $r\left({\psi }_{3,2}|\sqcap \right)=\left(2\zeta -7,2\zeta -5,2\zeta -1,0\right),$

$r\left({\psi }_{3,3}|\sqcap \right)=\left(2\zeta -6,2\zeta -6,2\zeta -2,0\right),$ $r\left({\psi }_{3,4}|\sqcap \right)=\left(2\zeta -5,2\zeta -5,2\zeta -3,0\right),$

$r\left({\psi }_{3,5}|\sqcap \right)=\left(2\zeta -4,2\zeta -4,2\zeta -4,0\right),$ $r\left({\psi }_{3,6}|\sqcap \right)=\left(2\zeta -3,2\zeta -3,2\zeta -5,0\right),$

$r\left({\psi }_{3,7}|\sqcap \right)=\left(2\zeta -2,2\zeta -2,2\zeta -6,0\right),$ $r\left({\psi }_{3,8}|\sqcap \right)=\left(2\zeta -1,2\zeta -1,2\zeta -5,0\right),$

$r\left({\psi }_{3,9}|\sqcap \right)=\left(2\zeta ,2\zeta ,2\zeta -4,0\right),$ $r\left({\psi }_{3,10}|\sqcap \right)=\left(2\zeta +1,2\zeta +1,2\zeta -3,0\right),$

$r\left({\psi }_{3,11}|\sqcap \right)=\left(2\zeta +2,2\zeta +2,2\zeta -2,0\right),r\left({\psi }_{3,12}|\sqcap \right)=\left(2\zeta +3,2\zeta +3,2\zeta -1,0\right),r\left({\psi }_{3,13}|\sqcap \right)=\left(2\zeta +3,2\zeta +4,2\zeta -1,0\right),r\left({\psi }_{3,14}|\sqcap \right)=\left(2\zeta +2,2\zeta +3,2\zeta +1,0\right),r\left({\psi }_{3,15}|\sqcap \right)=\left(2\zeta +3,2\zeta +4,2\zeta +4,0\right),r\left({\psi }_{3,16}|\sqcap \right)=\left(2\zeta +2,2\zeta +3,2\zeta +3,0\right),r\left({\psi }_{3,17}|\sqcap \right)=\left(2\zeta +3,2\zeta +4,2\zeta +4,0\right),r\left({\psi }_{3,18}|\sqcap \right)=\left(2\zeta +2,2\zeta +4,2\zeta +4,0\right),r\left({\psi }_{3,19}|\sqcap \right)=\left(2\zeta +1,2\zeta +3,2\zeta +3,0\right),r\left({\psi }_{3,20}|\sqcap \right)=\left(2\zeta ,2\zeta +2,2\zeta +4,0\right),$

$r\left({\psi }_{3,21}|\sqcap \right)=\left(2\zeta -1,2\zeta +1,2\zeta +3,0\right),r\left({\psi }_{3,22}|\sqcap \right)=\left(2\zeta -2,2\zeta ,2\zeta +4,0\right),$

$r\left({\psi }_{3,23}|\sqcap \right)=\left(2\zeta -3,2\zeta -1,2\zeta +3,0\right),$ $r\left({\psi }_{3,24}|\sqcap \right)=\left(2\zeta -4,2\zeta -2,2\zeta +2,0\right),$ $r\left({\psi }_{3,25}|\sqcap \right)=\left(2\zeta -5,2\zeta -3,2\zeta +1,0\right).$

Representations of the vertices of fourth cycle are given in the Tables 1 and 2.

Representations of the vertices of fifth cycle;

$r\left({\psi }_{5,1}|\sqcap \right)=\left(2\zeta -10,2\zeta -8,2\zeta ,0\right),r\left({\psi }_{5,2}|\sqcap \right)=\left(2\zeta -11,2\zeta -9,2\zeta -1,0\right),$

$r\left({\psi }_{5,3}|\sqcap \right)=\left(2\zeta -10,2\zeta -10,2\zeta -2,0\right),$ $r\left({\psi }_{5,4}|\sqcap \right)=\left(2\zeta -9,2\zeta -9,2\zeta -3,0\right),$

$r\left({\psi }_{5,5}|\sqcap \right)=\left(2\zeta -8,2\zeta -8,2\zeta -4,0\right),r\left({\psi }_{5,6}|\sqcap \right)=\left(2\zeta -7,2\zeta -7,2\zeta -5,0\right),$

$r\left({\psi }_{5,7}|\sqcap \right)=\left(2\zeta -6,2\zeta -6,2\zeta -6,0\right),r\left({\psi }_{5,8}|\sqcap \right)=\left(2\zeta -5,2\zeta -5,2\zeta -7,0\right),$

$r\left({\psi }_{5,9}|\sqcap \right)=\left(2\zeta -4,2\zeta -4,2\zeta -8,0\right),r\left({\psi }_{5,10}|\sqcap \right)=\left(2\zeta -3,2\zeta -3,2\zeta -9,0\right),$

$r\left({\psi }_{5,11}|\sqcap \right)=\left(2\zeta -2,2\zeta -2,2\zeta -10,0\right),r\left({\psi }_{5,12}|\sqcap \right)=\left(2\zeta -1,2\zeta -1,2\zeta -9,0\right),$

$r\left({\psi }_{5,13}|\sqcap \right)=\left(2\zeta ,2\zeta ,2\zeta +8,0\right),$ $r\left({\psi }_{5,14}|\sqcap \right)=\left(2\zeta +1,2\zeta +1,2\zeta -7,0\right),$

$r\left({\psi }_{5,15}|\sqcap \right)=\left(2\zeta +2,2\zeta +2,2\zeta -6,0\right),r\left({\psi }_{5,16}|\sqcap \right)=\left(2\zeta +3,2\zeta +3,2\zeta -5,0\right),$

$r\left({\psi }_{5,17}|\sqcap \right)=\left(2\zeta +4,2\zeta +4,2\zeta -4,0\right),r\left({\psi }_{5,18}|\sqcap \right)=\left(2\zeta +5,2\zeta +5,2\zeta -3,0\right),$

$r\left({\psi }_{5,19}|\sqcap \right)=\left(2\zeta +6,2\zeta +6,2\zeta -2,0\right),r\left({\psi }_{5,20}|\sqcap \right)=\left(2\zeta +7,2\zeta +7,2\zeta -1,0\right),$

$r\left({\psi }_{5,21}|\sqcap \right)=\left(2\zeta +7,2\zeta +8,2\zeta ,0\right),r\left({\psi }_{5,22}|\sqcap \right)=\left(2\zeta +6,2\zeta +7,2\zeta +1,0\right),$

$r\left({\psi }_{5,23}|\sqcap \right)=\left(2\zeta +7,2\zeta +8,2\zeta +2,0\right),r\left({\psi }_{5,24}|\sqcap \right)=\left(2\zeta +6,2\zeta +7,2\zeta +3,0\right),$

$r\left({\psi }_{5,25}|\sqcap \right)=\left(2\zeta +7,2\zeta +8,2\zeta +4,0\right),r\left({\psi }_{5,26}|\sqcap \right)=\left(2\zeta +6,2\zeta +7,2\zeta +5,0\right),$

$r\left({\psi }_{5,27}|\sqcap \right)=\left(2\zeta +7,2\zeta +8,2\zeta +6,0\right),$ $r\left({\psi }_{5,28}|\sqcap \right)=\left(2\zeta +6,2\zeta +7,2\zeta +7,0\right),$

$r\left({\psi }_{5,29}|\sqcap \right)=\left(2\zeta +7,2\zeta +8,2\zeta +8,0\right),r\left({\psi }_{5,30}|\sqcap \right)=\left(2\zeta +6,2\zeta +8,2\zeta +8,0\right),$

$r\left({\psi }_{5,31}|\sqcap \right)=\left(2\zeta +5,2\zeta +7,2\zeta +7,0\right),r\left({\psi }_{5,32}|\sqcap \right)=\left(2\zeta +4,2\zeta +6,2\zeta +8,0\right),$

$r\left({\psi }_{5,33}|\sqcap \right)=\left(2\zeta +3,2\zeta +5,2\zeta +7,0\right),r\left({\psi }_{5,34}|\sqcap \right)=\left(2\zeta +2,2\zeta +4,2\zeta +8,0\right),$

$r\left({\psi }_{5,35}|\sqcap \right)=\left(2\zeta +1,2\zeta +3,2\zeta +2,0\right),r\left({\psi }_{5,36}|\sqcap \right)=\left(2\zeta ,2\zeta +2,2\zeta +8,0\right),$

$r\left({\psi }_{5,37}|\sqcap \right)=\left(2\zeta -1,2\zeta +1,2\zeta +7,0\right),r\left({\psi }_{5,38}|\sqcap \right)=\left(2\zeta -2,2\zeta ,2\zeta +8,0\right),$

$r\left({\psi }_{5,39}|\sqcap \right)=\left(2\zeta -3,2\zeta -1,2\zeta +7,0\right),r\left({\psi }_{5,40}|\sqcap \right)=\left(2\zeta -4,2\zeta -2,2\zeta +6,0\right),$

$r\left({\psi }_{5,41}|\sqcap \right)=\left(2\zeta -5,2\zeta -3,2\zeta +5,0\right),r\left({\psi }_{5,42}|\sqcap \right)=\left(2\zeta +6,2\zeta -4,2\zeta +4,0\right),$

$r\left({\psi }_{5,43}|\sqcap \right)=\left(2\zeta -7,2\zeta -5,2\zeta +3,0\right),$ $r\left({\psi }_{5,44}|\sqcap \right)=\left(2\zeta -8,2\zeta -6,2\zeta +2,0\right),$

$r\left({\psi }_{5,45}|\sqcap \right)=\left(2\zeta -9,2\zeta -7,2\zeta +1,0\right).$

So the generalize set of vertices representations are given in the Table 3 and onward;

Representations of the vertices with $6\le ∊\le \zeta -1;$