Note on forgotten topological index of chemical structure in drugs

Development of drugs in medicine industry has more demand to determine the pharmacological, chemical and biological characteristics of the drugs. Many previous studies have pointed out that chemical and pharmacodynamics characteristics of drugs and their molecular structures are closely correlated [2, 4, 5, 9, 13].

In mathematical medicine, structure of the drug is represented as an undirected graph, where each vertex indicates an atom and each edge represents a chemical bond between the atoms.

Let G be a simple graph corresponding to a drug structure with vertex (atom) set V(G) and edge (bond) set E(G). The edge joining the vertices u and ν is denoted by uν. Thus, if uv ∊ E(G) then u and ν are adjacent in G. The degree of a vertex u, denoted by d(u), is the number of edges incident to u. Several topological indices such as Estrada index [1], Zagreb index [8], PI index [10], eccentric index [11], and Wiener index [12] have been introduced in the literature to study the chemical and pharmacological properties of molecules.

Furtula and Gutman [3] introduced the forgotten topological index of a graph G, also called as F−index, which is defined as

F(G)=∑u∈V(G)(d(u))3=∑uν∈E(G)[(d(u))2+(d(ν))2].$$\begin{array}{} \displaystyle F(G) = \mathop \Sigma \limits_{u \in V(G)} {(d(u))^3} = \mathop \Sigma \limits_{u\nu \in E(G)} [{(d(u))^2} + {(d(\nu ))^2}]. \end{array}$$

The study of forgotten topological index of the chemical structure in drugs is carried out in [6, 7]. Recently, Gao et al. [6] have obtained the forgotten topological index of some drug structures. The expression for the forgotten topological index of triangular benzenoid in their paper is erroneous. In this note we correct it. Also, we give the expression for forgotten topological index of graphene sheet which is more compact than the one obtained by Gao et al. [6].

The structure of the triangular benzenoid molecular graph T(n) is shown in Fig. 1. It has n(n + 1)/2 hexagons.

Fig. 1

In Gao et al. [6], the following result was obtained.

By means of structure analysis of T(n), we have |V₁| = 3n + 3 and |V₂| = n(n + 1) − 2. Therefore, by the definition of forgotten topological index, we get

F(T(n))=∑u∈V(T(n))(d(u))3=∑u∈V1(d(u))3+∑u∈V2(d(u))3=(3n+3)(8)+(n(n+1)−2)(27)=27n2+51n−30.$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {F(T(n)){\rm{\;}}}&{ = {\rm{\;}}\mathop \sum \limits_{u \in V(T(n))} {{(d(u))}^3}}\\ {}&{ = {\rm{\;}}\mathop \sum \limits_{u \in {V_1}} {{(d(u))}^3} + \mathop \sum \limits_{u \in {V_2}} {{(d(u))}^3}}\\ {}&{ = {\rm{\;}}(3n + 3)(8) + (n(n + 1) - 2)(27)}\\ {}&{ = {\rm{\;}}27{n^2} + 51n - 30.} \end{array} \end{array}$$

Graphene is a planar sheet of carbon atoms that is densely placed in a honeycomb crystal lattice. It is the main element of certain carbon allotropes. The structure of graphene G(m,n) with n rows and m columns is shown in Fig. 2. It has mn hexagons.

Fig. 2

In Gao et al. [6], the following result was obtained.

Table 1

Number of edges in each row of G(m,n).

Using the data of Table 1, we can easily write |E₁| = n + 4, |E₂| = 4m + 2n − 4 and |E₃| = 3mn − 2m − n − 1. Therefore, by the definition of forgotten topological index, we obtain

F(G(m,n))=∑uν∈E(G(m,n))[(d(u))2+(d(ν))2]=∑uν∈E1[(d(u))2+(d(ν))2]+∑uν∈E2[(d(u))2+(d(ν))2]+∑uν∈E3[(d(u))2+(d(ν))2]=(n+4)(8)+(4m+2n−4)(13)+(3mn−2m−n−1)(18)=54mn+16m+16n−38.$$\begin{array}{*{20}{l}} {F(G(m,n))} & { = \sum\limits_{u\nu \in E(G(m,n))} [{{(d(u))}^2} + {{(d(\nu ))}^2}]} \\ {} & { = \sum\limits_{u\nu \in {E_1}} [{{(d(u))}^2} + {{(d(\nu ))}^2}] + \sum\limits_{u\nu \in {E_2}} [{{(d(u))}^2} + {{(d(\nu ))}^2}] + \sum\limits_{u\nu \in {E_3}} [{{(d(u))}^2} + {{(d(\nu ))}^2}]} \\ {}& { = (n + 4)(8) + (4m + 2n - 4)(13) + (3mn - 2m - n - 1)(18)} \\ {}& { = 54mn + 16m + 16n - 38.}\end{array}$$

In this note, we have given the correct expression for the forgotten topological index of triangular benzenoid. Also, a simple expression for the forgotten topological index of graphene sheet is derived.