Topological Indices, and Applications of Graph TheoryView this Special Issue
Research Article | Open Access
Suha Wazzan, Anwar Saleh, "Locating and Multiplicative Locating Indices of Graphs with QSPR Analysis", Journal of Mathematics, vol. 2021, Article ID 5516321, 15 pages, 2021. https://doi.org/10.1155/2021/5516321
Locating and Multiplicative Locating Indices of Graphs with QSPR Analysis
In this paper, by introducing a new version of locating indices called multiplicative locating indices, we compute exact values of these indices on well-known families of graphs and graphs obtained by some operations. Also, we determine the importance of locating and multiplicative locating indices of hexane and its isomers. Furthermore, we show that locating indices actually have a reasonable correlation using linear regression with physico-chemical characteristics such as enthalpy, melting point, and boiling point. This approximation can be extended into several chemical compounds.
As an example of a molecular descriptor, a topological graph index is defined as a mathematical formula which is applied to any graph that models some molecular structure. These indices make analyzing mathematical values and examining certain molecules physico-chemical properties more feasible and efficient by enabling us to bypass costly and lengthy laboratory experiments. The role of molecular descriptors is well established in mathematical chemistry. They include but are not limited to or quantitative structure-property relationship. There are various topological indices in the literature, and many of them have broad applications in chemistry. The structural properties of the graphs employed in the calculations can be used to classify them. For instance, the Zagreb type indices are computed using the degrees of vertices in a graph. They helped to compare some alkane isomers boiling points and have aided in the discovery, along with other indices, of a few thousand topological graph indices enrolled in the chemical data bases. In fact there has been a rapidly increasing interest of this topic, and thus topological graph indices have been studied worldwide by both mathematicians and chemists (see [1–8]). The most widely known topological indices are the first and second Zagreb indices, which have been introduced by Gutman and Trinajstic in , and defined as and , respectively. Actually, several new versions of the Zagreb indices have been established for similar purposes (cf. [10–17]).
Different topological indices for some chemical compounds such as “aspirin” and the anticancer drug “carbidopa” have been studied in detail by Wazzan (cf. [18, 19]). Moreover, in a recent work, Wazzan et al. (see ) introduced novel topological indices called the first and second locating indices. To do that, the authors used the locating matrix over a graph (cf. ). Let be a connected graph with the vertex set . A locating function of denoted by is a function such that , where is the distance between the vertices and in . The vector is called the locating vector corresponding to the vertex , where is actually the dot product of the vectors and in the integers space such that is adjacent to . In the present paper, as a next step of the work in , we introduce the first and second multiplicative locating indices for a connected graph as in the following definition.
Definition 1. For a connected graph with an edge set and vertex set , the first and second multiplicative locating indices are defined as follows:respectively.
In this paper, we only consider simple graphs with no multiple edges. For the terminologies, we may recommend citation  to readers.
2. Certain Values of Multiplicative Locating Indices
In this section, by considering Definition 1, we will determine the first and second multiplicative locating indices for some special graphs such as , , , , and , and also we will compute the same indices for the graph such that is obtained by joining two graphs and (notationally ), where and are connected with diameter 2. In particular, we will assume that as - and -free graphs.
Theorem 1. Let be the complete graph with . Then,(1)(2)
Proof. (i)Let be the complete graph with and let , and for each vertex , we let is the locating vector associated with the vertex . Then, such that and all the other components are equal to 1. Hence, . However, the total amount of vertices in is vertices, and so, .(ii)For any arbitrary locating vectors and , where , we gain . Therefore, .
Theorem 2. Let , where . Then,(1)(2)
Proof. We identify the adjacent vertices and of , for all and . Then, the locating vectors of are given byHere, for any , we have and for any , therefore, . Therefore,Similarly, for any two locating vertices in where adjacent to , . Hence, .
Corollary 1. Let , where . Then,
Corollary 2. Let be any star graph . Then,
Theorem 3. For an even integer , let . Then,
Proof. By identifying the vertices of the cycle as in the anticlockwise direction, we obtainand hence . It is straightforward to see that each has equivalent components but in different locations; hence, each has the same sum as the form ofTherefore, . By symmetry,which gives .
Theorem 4. Let with an odd number of vertices . Then,
Proof. Following the steps in the proof of Theorem 4, we getand with a simple calculation, one can obtainwhich implies . Further, by the symmetry,Hence, , as required.
Theorem 5. Let be wheel graph with vertices such that . Then,
Proof. Let with vertices. Suppose that the vertices are labeling in the anticlockwise direction where the center of the wheel is labeled . Hence, we getTherefore, for each corresponding locating vector with the vertex , we have and . So, . For , by considering the same labeling as previously, we getHere, the permutation components in each vector where are . Hence, it is straightforward to notice that any two adjacent vertices and satisfy and for . Therefore, . Hence, the result is obtained.
Theorem 6. For any path with vertices,
Proof. Assume that is the path with vertices. Suppose that the locating function is constructed by identifying the vertices as from left to right. Hence, the corresponding vectors for each vertex are given as in following:A straightforwardly calculation implies thatFor the other case ,So, we getTherefore, is obtained as required in the statement of theorem.
In the following result, we will give our attention to the join of graphs and for computing multiplicative locating indices.
Theorem 7. Let such that and are both connected graphs, where and have edges; vertices and edges vertices, respectively. Then,
Proof. Let be as in the statement of theorem. Let us label the vertices of the graph aswhere and . In addition, suppose thatwhich is the locating vector associated with the vertex . Then, . Similarly, for any vertex , the locating vector corresponding to is given bySo, . Therefore, by the above equalities on and , we obtain as required in the theorem.
Theorem 8. Suppose that and are connected graphs having diameter 2. Let such that is a - or -free graph. Assume that has edges and vertices while has edges and vertices. Then,where
Proof. Under the assumptions on as in the statement of the theorem, the partition sets edges are defined byHence, is expressed asFor any two adjacent vertices to obtain , we assume that the first two vertices as follows:Since and are - or -free graph, for any two vertices and in , we can obtainwhich implies .
With the same way of calculation, we get . Now, to achieve the computation of , let us take and . Thus,As a result, we getand so . Then, by all above calculations, we finally get , whereHence, the result is obtained.
Theorem 9. Let be a book graph with vertices. Then,
Proof. Let us label the vertices as in Figure 1. So, we haveConsidering the components of the locating vectors of the book graph, we get , and for , we have . Hence, . Similarly, we have , and for any , , and in the same way, . Hence, .
3. Multiplicative Locating Indices of Firefly Graphs
A firefly graph (, , and ) is a graph of order that consists of triangles, pendant paths of length 2, and pendent edges that are sharing a common vertex . Let be the set of all firefly graphs . Note that contains the stars , stretched stars , friendship graphs , and butterfly graphs .
In the following result, the first and second multiplicative locating indices for the firefly graph are calculated. To simplify the calculations, let us denote by .
Theorem 10. Let () be a firefly graph of order . Then,
Proof. Suppose that () is a firefly graph of vertices. Let us label the vertices of the graph (see Figure 2) with clockwise direction.
So, in the setwhere is the center vertex of the firefly graph, is the vertices of the triangles, is the vertices of the pendent edges, is the first vertices of the pendent paths, and be the second vertices of the pendent paths. Therefore, we obtain the corresponding vectors for each vertex where as follows: