Abstract
Vukičević and Gasperov introduced the concept of 148 discrete Adriatic indices in 2010. These indices showed good predictive properties against the testing sets of the International Academy of Mathematical Chemistry. Among these indices, twenty indices were taken as beneficial predictors of physicochemical properties. The inverse sum indeg index denoted by of is a notable predictor of total surface area for octane isomers and is presented as , where represents the degree of . In this paper, we determine sharp bounds for ISI index of graph operations, including the Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, union of graphs, double graph, and strong double graph.
1. Introduction
Let be a connected and simple graph whose vertex and edge sets are and , respectively. The order and size of are the cardinalities of and , respectively. The degree formula of is the cardinality of linked vertices to in and represented by . The largest (or smallest) degree of is the degree of a vertex of with the greatest (or least) number of edges incident to it and represented by (or ).
A molecular descriptor is a numerical parameter of a graph that distinguished its topology. In organic chemistry, topological descriptors have investigated many applications in pharmaceutical drug design, QSAR/QSPR study, chemical documentation, and isomer discrimination. Some of these topological indices are Wiener index, Zagreb indices, Szeged index, and Randić index. The set of 148 discrete Adriatic descriptors [1] have been defined in 2010. These descriptors showed well predictive characteristics on the testing sets given by International Academy of Mathematical Chemistry. Twenty of these descriptors were taken as noteworthy predictors of physicochemical properties. One such index is inverse sum indeg index, denoted by , of that was investigated in [1] as a noteworthy predictor of total surface area for octane isomers and is presented as follows:
Sedlar et al. [2] investigated graph-theoretical characteristics of ISI index. Falahati-Nezhad et al. [3] computed some sharp bounds of inverse sum indeg (ISI) index.
The Zagreb indices of are presented by Gutman and Trinajstić [4] as follows:
Let be -vertex and be -vertex graphs with size and , respectively. The Cartesian product , whose vertex set is and and are adjacent when and or and , is a graph. The order and size of are and , respectively. The degree formula for is .
The tensor product , whose set of vertices is and and are linked when and , is a graph. The order and size of are and , respectively. The degree formula for in is .
The strong product , whose vertex set and edge set are and , respectively, is a graph. The order and size of are and , respectively. The degree formula for in is .
The composition , whose vertex set and and are linked when or and , is a graph. The order and size of are and , respectively. The degree formula for in is .
The disjunction , whose vertex set is and and are linked when or is a graph. The order and size of are and , respectively. The degree formula for in is .
The symmetric difference is a graph with vertex set and whenever or but not both. The order and size of are and , respectively. The degree formula for is .
Let be all vertex disjoint graphs. Then, their join is a graph whose vertex set is and edge set is together with the edges linking and , and so on and . The degree formula of is , and .
The corona product is acquired by taking as a single copy and copies of and by linking -th vertex of to every vertex of r-th copy of , where . The graph has size and order and , respectively. The degree formula of is
The Indu–Bala product is obtained from two disjoint copies of by linking the corresponding vertices of two copies of . The order and size of are and , respectively. The degree of is
The double graph is acquired by taking original edge set of two copies and of and linking each vertex in with the linked vertices of corresponding vertex in . The strong double graph is acquired by taking two copies of and of and linking each vertex in with closed neighborhood of corresponding vertex in .
Figure 1 depicts some graph operations. For more details on these graph operations, see [5–14]. Also, we refer some recent articles [15–19] on different kinds of descriptors.It is an important and well-reputed problem to study and explore the molecular topological descriptors of the graph operations in terms of the original graphs, say and , and this also helps to explore the physicochemical properties of the complex chemical structures which arise from these graph operations. The upper and lower bounds of any molecular descriptors are the important information related to a chemical graph. They determine the approximate possible range of the invariant in the form of molecular structural parameters. There are some bounds already available for the inverse sum indeg (ISI) index regarding the number of pendant vertices, size, radius, smallest and largest vertex degrees, and smallest nonpendent vertex degree of a graph computed in [3]. The objective of this article is to determine the bounds for inverse sum indeg index of some graph operations including Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, union of graphs, double graph, and strong double graph in the form of original graphs, say and .
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2. Applications of Graph Theory Concept and Topological Indices in Chemistry
In 1936, Hosoya introduced the concept of graph terminologies in chemistry and provided a modeling for molecules. This modeling contents lead to predict the chemical properties of molecules, easy classification of chemical compounds, computer simulations, and computer-assisted design of new chemical compounds. As in current century, chemists manipulate graphs on a daily basis using Table 1 terminologies for recent development in their research.
Graph hypothesis had investigated an interesting exercise around in research. Compound graph speculation has provided a collection of beneficial indices, for instance, topological indices. The Zagreb indices are the topological indices that are correlated to a substantial computation of fabricated characteristics of the particles and have been investigated parallel to establishing the Kovats constants and limit of the particles [20]. The hyper Zagreb descriptor has a strong bound between the security of direct dendrimers besides the expanded medication stores and for establishing the strain criticalness of cyclo alkanes [21]. To connect with various physico-mix characteristics, Zagreb indices have required deep control upon the essentialness of the dendrimers [22]. The Zagreb polynomials were determined to happen for computation of the -electron imperativeness of the particles inside specific brutal verbalizations [23, 24].
3. Inverse Sum Indeg Index of Graph Operations
In this section, we compute the inverse sum indeg index of the Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, double graph, and strong double graph. The relation between largest and smallest degree of to the degree of is as follows:
In the upcoming theorem, we calculate the bounds for inverse sum indeg (ISI) index of Cartesian product.
Theorem 1. Let and be two graphs. Then,The equalities hold if and only if and are regular.
Proof. Using the degree formula for a vertex of in equation (1),Similarly, we can evaluateThe above equalities hold if and only if factor graphs are regular.
In the next theorem, we calculate the bounds for ISI index of tensor product of and .
Theorem 2. Let and be two graphs. Then,The above equalities hold if and only if both graphs are regular.
Proof. Using the degree formula for a vertex in tensor product of graphs in (1),See Theorem 2.1 in [25]. Similarly, we can computeThe above equalities hold if and only if factor graphs are regular.
We derive the bounds of inverse sum indeg (ISI) index of in the upcoming theorem.
Theorem 3. Let and be two graphs. Then,The equalities hold if and only if both graphs are regular.
Proof. Using the degree formula of a vertex in strong product of graphs in (1),In a similarly way,The above equalities satisfy if and only if factor graphs are regular.
In the upcoming theorem, we evaluate the bounds for inverse sum indeg (ISI) index of .
Theorem 4. Let and be two graphs. Then,The equalities hold if and only if both graphs are regular.
Proof. Using the degree formula of an element of in (1),In a similar way,The above equalities hold if and only if factor graphs are regular.
In the following theorem, we present the bounds for inverse sum indeg (ISI) index of disjunction of and .
Theorem 5. Let and be two graphs. Then, The equalities hold when factor graphs are regular.
Proof. Using the degree formula of an element of in (1),Similarly, we computeThe above equalities hold when both graphs are regular.
Next, we derive the bounds of inverse sum indeg (ISI) index of .
Theorem 6. Let and be two graphs. Then,The equalities hold if and only if factor graphs are regular.
Proof. Using the degree formula of a vertex of in (1),Similarly,The above equalities hold if and only if both graphs are regular.
Next, we evaluate the bounds of inverse sum indeg (ISI) index of join of graphs.
Theorem 7. Let . Then,The equalities hold if and only if , for , are regular graphs.
Proof. We assume that , for and . By using the degree formula of a vertex in given in (1),SimilarlyThe above equalities hold if and only if , , are regular.
In the following theorem, we calculate the bounds for ISI index of .
Theorem 8. Let and be -vertex and -vertex graphs. Then,The equalities hold if and only if both graphs are regular.
Proof. Using the degree formula of a vertex in corona product in (1),From equation (2), we obtainSimilarly, we calculateThe above equalities hold only when and are regular graphs.
Next, we evaluate the bounds for inverse sum indeg (ISI) index of Indu–Bala product.
Theorem 9. Let and be -vertex and -vertex graphs. Then,The equalities hold only when and are regular.
Proof. Using the degree formula of a vertex in Indu–Bala product in (1),Using equation (2), then we haveSimilarly, we calculateThe equalities hold only when and are regular graphs.
In the next theorem, we find the inverse sum indeg (ISI) index of double graph.
Theorem 10. Let be a -vertex graph. Then,
Proof. Using the degree formula of a vertex in in equation (1), we acquireIn the upcoming theorem, we calculate the bounds for inverse sum indeg (ISI) index of strong double graph.
Theorem 11. Let be an -vertex graph. Then,The equalities hold only when is regular.
Proof. Using the degree formula of a vertex in in (1),Similarly, we computeThe above equalities hold only when is a regular graph.
4. Conclusion
In this paper, some graph operations including different products, differences, union of graphs, double graph, and strong double graph are studied. In particular, we have found the sharp bounds for inverse sum indeg (ISI) index of these operations of graphs. The investigation related to other significant predictors is still open.
Data Availability
All kinds of data and materials, used to compute the results, are provided in Section 1.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project was sponsored by the Deanship of Scientific Research under Nasher Proposal No. 216006, King Faisal University.