Distance-Based Topological Descriptors on Ternary Hypertree Networks

School of Big Data and Artificial Intelligence, Anhui Xinhua University, 230088 Hefei, China Department of Mathematics, Loyola College (Affliated to University of Madras), Chennai, India Department of Mathematics, St. Joseph’s College, Bangalore, India Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia


Introduction
A connected graph having order n and size n − 1 is termed as a tree that contains no cycle. In computer science, trees are designed as data structures. Trees are helpful to store data information in a hierarchical manner and provide insertion and deletion of data. ey are also useful in manipulating hierarchical data, making it easy to search information and aid in multistage decision making. One of the basic tree structures which have many applications in the field of computer science is the rooted tree [1,2]. Rooted tree is a tree that has a root node from where the children arise. e root node is called the parent node [3,4]. A binary tree is a rooted tree in which every vertex has at the most two children and each child of a vertex is assigned as its left child or right child [5]. A complete binary tree is a rooted tree in which every node has two children-a right child and a left child. Ternary tree which has at the most three children 3x − 1, 3x, 3x + 1, where x ∈ Z is a root node, is a rooted tree. Ternary tree is introduced by Barning, a Dutch mathematician in Reference [6]. It is a tool for the ternary search tree which can be used in spell check and as a database when indexing several nonkey fields. In a complete ternary tree, every node has exactly three children.
Hypertree of dimension n is a basic skeleton of complete binary tree, i.e., the vertex x has exactly two children 2x and 2x + 1, where x ∈ 2 n− 1 − 1, and the vertices on the same level are connected by a horizontal edge with a label difference of 2 i− 2 ; 2 ≤ i ≤ n. e hypertree is an interconnection network which has minimum average distance which results in an efficient multicomputer system [7]. It has an excellent combination of characteristics of the hypercube and the binary tree. Recursive hypertrees are modelled as biological networks such as dendrimers [8][9][10]. e branching of biological networks is not restricted to two branches. With this motivation, we introduce the concept of ternary hypertree. Ternary hypertrees can be modelled as biological networks for protein interactions and to analyze the spread of diseases. e structure of the ternary hypertree is a combination of a complete ternary tree and hypertree. It is a spanning subgraph of the complete ternary tree. We denote ternary hypertree with dimension n as THT(n). e level of root node is 1. e root node gives rise to three children, which is at level 2. We label the root node as 1 and their children as 2, 3, 4. Likewise, if the node is labelled as x; then, the children are labelled as 3x − 1, 3x, 3x + 1 where x ∈ [3 n− 1 − 1/2]. At each level i; 1 ≤ i ≤ n of the ternary hypertree of dimension n has 3 i− 1 nodes. For a ternary hypertree of dimension n, the network has n levels. ere are horizontal edges in the level i, 2 ≤ i ≤ n connecting the nodes with a label difference of 3 i− 2 along the complete ternary tree structure. See Figure 1.
Ternary hypertree consists of (3 n − 1/2) nodes and (3 n − 3) edges. e vertex connectivity is 4 and edge connectivity is 3. Ternary hypertree of dimension n has a diameter of 2n − 3. Also, it is not a regular network. THT(n); n ≥ 3 is nonplanar, i.e., it cannot be embedded in a plane and non-Hamiltonian where every vertex can be visited more than once.
Real-life problems can be converted to graphical representations using mathematical modelling, especially in the field of biology [11][12][13][14]. Networks helps in analysing various health problems by modelling the spread of diseases [15][16][17]. Topological indices are numeric invariants showing a correlation between the subatomic structure and its physical (as well as chemical) properties [18,19]. us, it characterises the topology of a graph [20,21]. Topological indices analyse the physical, chemical, and biological characteristics of a synthetic framework [22,23]. Topological indices are essential in the field of chemistry and pharmacology, notably in nanomedicine. It helps in the study of the properties of networks. ese descriptors are used in measuring irregularity, connectivity, centrality, and peripherality in networks [24]. Topological indices for various networks have been studied in recent years [25][26][27][28].
Computing the topological indices helps in anatomising the properties of the biological network. In the next section, we have discussed some terminologies and two types of topological descriptors (distance-based and degree-based descriptors) of the ternary hypertree are derived and are graphically represented. Section 3 concludes the paper with discussion on the possible applications of ternary hypertree.

Topological Indices
e graph Ω considered in the paper is a simple connected graph. d(x, y) is used to represent the distance between x and y and is the length of the shortest path connecting the vertices, x and y.
e cardinality of collection of adjacent vertices of x is termed as the degree of a vertex x, it is denoted by d x [29,30]. Neighbourhood of a vertex, x is represented by N(x) and is defined as follows: and We denote the cardinality of N x (xy|Ω) and M x (xy|Ω) as n x (xy|Ω) and m x (xy|Ω), respectively.
Let (w v , s v ) be the vertex weight and vertex strength and let (e w , s e ) be the edge weight and edge strength. e notion of strength-weighted graph We refer to References [32][33][34] for the distance-based topological indices. e formulas of these indices for strength-weighted graph Ω sw are given in Table 1 and the degree-based formulas of topological indices of graph Ω are illustrated in Table 2.
In this paper, we consider If the distance of any two vertices in H, a subgraph of a graph of Ω, lies in the same subgraph, then the subgraph H is called convex. For Ω, Djoković-Winkler's relation Θ on E(Ω), References [41,42] can be expressed as follows: if Θ is an equivalence relation in case of partial cubes. Θ partitioned E(Ω) into convex cuts. Θ * (a transitive closure) is an equivalence relation. e edges partition into Θ * classes and let F i ; 1 ≤ i ≤ k is the Θ * partition set of E(Ω). Using Θ * relation, we can find the topological indices of any graph [31,40,[43][44][45]. For any i ∈ [k], the quotient Ω/F i graph in which vertex set belongs to the components of Ω − F i and x, y ∈ Ω/F i are adjacent in Ω/F i if xy ∈ E(Ω), where x ∈ C 1 , y ∈ C 2 and where C 1 , C 2 are components. A partition X � X 1 , X 2 , . . . , X r of E(Ω) is coarser than Y � Y 1 , Y 2 , . . . , Y s if X i is the union of one or more sets in Y. To study about the Wiener index, see References [46][47][48]. We have used eorem 2.1 and the technique in Reference [48], reduction of original graph Ω into quotient graphs and further into reduced graphs, to compute the Wiener index of ternary hypertree. To compute other distance-based topological indices of ternary hypertree, we use eorem 1. [49,50]. "For a connected strengthweighted graphG sw � (G, (w v , s v ), s e ), letE � E 1 , E 2 , . . . , E k be a partition ofE(G)coarser thanF. LetX � W, Sz v , Sz e , Sz ev , Mo, Mo e , Mo t , PI. en,