Abstract
Suppose is a simple graph, then its eccentric harmonic index is defined as the sum of the terms for the edges , where is the eccentricity of the vertex of the graph . We symbolize the eccentric harmonic index (EHI) as . In this article, we determine for the Cartesian product (CP) of particularly chosen graphs. Lower bounds for of the CP of the two graphs are established. The formulas of EHI for the Hamming and Hypercube graphs are obtained. These obtained formulas can be used in QSAR and QSPR studies to get a better understanding of their applications in mathematical chemistry.
1. Introduction
Authors Throughout this study, we assume that all graphs are simple, connected as well as finite. A graph is known as a simple graph if it has no loops, no directed, and multiple edges. Whereas and depict the cardinality of the graph . The open neighborhood of for in the graph , is symbolized by . consists of all nodes which are adjacent to and the closed neighborhood of is . The degree of a node in is . The graph is said to be -regular graph if for all . The shortest distance between any two connecting nodes is denoted by for a graph . The eccentricity of a node is . The radius of is , and the diameter of is . Hence, for every . In a connected graph , a node is a central node if , while a node in a connected graph is the peripheral node if . It is important to remember that for a self-centered graph , the condition must be true.
If the graph is regular under the constraint , then is familiarized as a regular self-centered graph. We depict the eccentricity of a node by . Denoted by the path, complete bipartite, cycle, complete, and star graphs, respectively. The reference includes all terminologies and definitions.
2. Historical and Preliminary Overview
We have a set of particular fixed parameters (invariants) which have no changes for isomorphic graphs and give us some information about various factors. They are familiarized with the term “topological descriptors.” These topological descriptors (TDs) are very useful in solving many everyday life problems. They have a vital role in the graphical construction of chemical compounds. TDs have many uses in various areas such as the molecular graph theory because of their benefits over the existing experimental methods. The pharmaceutical industries paid their keen roles towards the increased day-by-day interest in TDs because of the need of human beings to low the expenditures involved in the synthesis, in vivo or in vitro clinical testing of the latest medicinal compounds. TDs can be categorized into the following three subclasses according to their properties:(i)Formulated by means of matrices(ii)Formulated by means of node degrees(iii)Formulated by means of distances
The most commonly used TDs [1, 2] are the geometric-arithmetic index, the first and second Zagreb indices, the harmonic index, the first and second multiplicative Zagreb indices, the Narumi–Katayama index, the sum-connectivity index, and the atom-bond connectivity index.
During 1972–1975, Trinajstic and Gutman formulated and elaborated [3, 4] the 1st and 2nd Zagreb indices and determined by
The harmonic index H(ρ) of a graph ρ was developed by the mathematician Fajtlowicz [5–10], and defined by
For details about H(ρ), the authors' advice see [11–14].
In 2000, the connective eccentric index Cξ(ρ) was defined by Gupta et al. [15], as follows:
The eccentric harmonic index He(ρ) of a graph ρ was determined by Sowaity et al. [16], and formulated by
Pavithra et al. [17] computed the EHI for the subdivision of particular graphs. Motivated by this, we compute the EHI for the CP of particular graphs.
Definition 1. For the given graphs and , their CP [18] is symbolized by and determined as the graph on the vertex set , and vertices and of are connected by an edge iff either or . We know that the CP of graphs is associative as well as commutative. , the distance between any two nodes and in is given by . The eccentricity of the vertex is symbolized by .
Definition 2. (1)A Hypercube graph is the CP of copies of (2)The Hamming graph is equivalent to the CP of copies of the complete graph
Definition 3. The ladder graph is defined as the CP of the path and , i.e., .
Definition 4. The circular ladder graph is formulated as the CP . The following results are used in proving the main and the secondary results:
Observation 1. Let m, n ∈ Z+, m, n ≥ 2. Then, m + n ≤ mn with equality holds iff m = n = 2.
Lemma 1 (see [19]). Suppose and are two disjoint graphs with vertices, respectively. Then, for any node .
Theorem 1. Let be a graph that is self-centered [16] having order and size , then .
Corollary 1 (see [16]). Let be a cycle, then its EHI is .
3. Eccentric Harmonic Index for the Cartesian Product of Graphs
We derive the EHI for the CP of some famous graphs which have a good application in Chemistry, Electronics, Electrical, and Wireless communication areas. Eccentric harmonic index for the Hamming and Hypercube Graphs are obtained.
Corollary 2. Suppose and are two disjoint graphs having vertices, respectively. Then, , and the above equality holds iff and both are complete graphs.
Proof. Suppose and are two disjoint graphs with nodes and edges, respectively. Then by definition of of a graph and Lemma 1, we getFrom Observation 1, we getTo show equality, by using inequality (7), we get that must be a self-entered graph. Also, by inequality (6) we get the equality iff and , which holds if and only if and .
Thus, the result is as follows:
Corollary 3. Suppose and are two disjoint self-centered graphs having vertices, edges, and diameters, respectively. Then,with equality holds iff and are both complete graphs.
Proof. Suppose and are two disjoint graphs with vertices, edges, and diameters, respectively. Then by Corollary 2, we getBy using Theorem 1, we getAssume that the size of . Then, clearly . Thus, by employing , we can rewrite (8) as
Corollary 4. Let and be two disjoint complete graphs with vertices, respectively. Then, .
Proof. Let and be two disjoint complete graphs with vertices, respectively. Then by Corollary 3, we getSince and are both self-centered graphs with , then
Theorem 2. Suppose are -disjoint graphs with orders , respectively. Then, the eccentricity of any vertex is
Proof. Suppose are disjoint graphs with orders , respectively. Assume that t = 2. Then by Lemma 1, we getSuppose it is true for , i.e.,Then, for , from Lemma 1 and since the Cartesian product is associative, we getHence,
Theorem 3. Let be disjoint complete graphs with orders , respectively. Then,where .
Proof. Let be disjoint complete graphs with orders , respectively. Then by the definition of of a graph , we getBy Theorem 2, we getLet . Then,
Corollary 5. The eccentric harmonic index of the Hamming graph is
Proof. Let be the Hamming graph. Then by substituting and in (18), we get
Corollary 6. The eccentric harmonic index of the Hypercube graph is .
Proof. Let be the Hypercube graph. Then by substituting in (22), we get
Theorem 4. The eccentric harmonic index for the ladder graph is .
Proof. Let be the ladder graph. Then by the definition of , we getBy Lemma 1,Moreover,By substituting (26) and (27) in (25), we get
Theorem 5. The eccentric harmonic index for the circular ladder graph is .
Proof. By Lemma 1,Thus,By using Corollary 1, we getHence,Furthermore, we haveBy substituting (33) and (34) in (29), we get
4. Importance of the Work
The EHI has recently been projected in a comparable way by swapping the vertex degrees with the vertex eccentricities. One of the main topics in the chemical graph theory is to study how certain invariants of product graphs are related to the corresponding invariants of their components. Due to this, here we investigate the behavior of the eccentric version of the harmonic index under various families of graph products and apply the derived results to some graphs of chemical and general interest.
5. Hypothesis and Limitations of the Industrialized Formulas
There are some partial limitations of this work which are as follows:(1)These formulas are incompletely characterized(2)There is limited knowledge of chemistry for the molecular structures of Hamming and Hypercube graphs(3)We are constrained to hypothesis nth formula to calculate topological indices which can be utilized to interlink with the molecular compound
6. Conclusion
We have discussed and developed some important formulas. These obtained formulas of EHI for the Hamming and Hypercube graphs can be correlated with the physical as well as biological properties to get a better understanding of the topology of given molecular compounds.
6.1. Future Work
It is an open problem to apply the eccentric harmonic index in other fields such as Physics and Computer science.
Data Availability
All data used are included inside the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.