Szeged-like topological indices and the efficacy of the cut method: The case of melem structures∗

The Szeged index is a bond-additive topological descriptor that quantifies each bond’s terminal atoms based on their closeness sets which is measured by multiplying the number of atoms in the closeness sets. Based on the high correlation between the Szeged index and physico-chemical properties of chemical compounds, Szeged-like indices have been proposed by considering closeness sets with bond counts and other mathematical operations like addition and subtraction. As there are many ways to compute the Szeged-like indices, the cut method is predominantly used due to its complexity compared to other approaches based on algorithms and interpolations. Yet, we here analyze the usefulness of the cut method in the case of melem structures and find that it is less effective when the size and shape of the cavities change in the structures.


Introduction
Chemical graph theory is applied extensively in the field of quantitative structural activity and property relationships (QSAR/QSPR), which has great importance in modern chemistry, pharmocology, chemometrics, toxicology, and so on [4,6,19]. This has led to the emergence of various molecular descriptors, predominantly the topological indices, for the prediction of physicochemical properties of compounds, as research shows that the properties of compounds are intimately related to their underlying topological nature. The need to represent a molecular structure by a single number arises from the fact that most molecular properties are recorded as single numbers. Therefore, QSPR modelling reduces to a correlation between the two sets of numbers via an algebraic expression [14]. During the past decades, various topological indices have been defined and studied for their development in the study of quantitative structure-property relations [1,4,6,7,10,11,15,21].
Wiener's pioneering work in predicting the boiling point of parrafin using the path number broadened the scope of QSAR/QSPR studies by predicting various correlations between the physicochemical properties of chemical compounds. Since then, many indices were introduced based on the distance, degree, and bond additive invariants of a graph. Gutman introduced the vertex variant of the Szeged index based on the bond-additive structural invariant that was used to ease the computation of the Wiener indices for trees, and since its existence, a lot of research has been devoted towards its study as a useful molecular topological descriptor. Various physicochemical properties of organic compounds such as molecular volume, boiling point, vapour pressure, molar volume, van der Waals volume, proton-ligand formation constants and so on were modelled using Szeged index (see [11]). Consequently, several variations of the Szeged index were introduced for possible applications in QSAR/QSPR studies [7,16]. In particular, the PI index has fairly good structural selectivity and correlation ability [10], while the Mostar index provides a quantitative measure of distance nonbalancedness as well as a measure of the global peripherality of molecular structures [1,3,5].
To formally define the Szeged-like indices, we need to recall a few graph theoretical notions. The open neighborhood N G (v) of a simple connected graph G, consisting of a vertex set V (G) and an edge set E(G), is the set of vertices adjacent to v. Its cardinality is the degree of v and denoted by d G (v). For an edge e = xy in E(G), the weighted sum/product is based on the degree of the end-vertices and is given by the distance d G (x, y) between them is the number of edges in a shortest path from the vertex x to y. The shortest distance between the vertex x and the edge f = uv ∈ E(G) is defined as d G (x, f ) = min{d G (x, u), d G (x, v)}. The cardinalities of the closeness sets of an edge f = uv are defined in the following.
Based on the above notations, we now define the Szeged-like topological indices in the following form [3,18], whereas the reductions can be accomplished by assigning appropriate values to k, w, p and employing mathematical operations at •.

The cut method
The cut method was introduced in order to simplify the computation of topological indices and to derive closed formulas for chemically important families of graphs. The method is based on the Djoković-Winkler relation Θ which is defined as follows.
The relation Θ is reflexive and symmetric, but not transitive in general whereas the transitive closure Θ * forms an equivalence relation thereby enabling the Θ * -partition of the edge set E(G) as E 1 , . . . , E p . These classes split each of the graphs G − E i into two or more smaller components. The quotient graph G/E i is defined as a graph in which the vertices are the connected components of G − E i , and two components A 1 and A 2 are linked by an edge if there exists an edge xy ∈ E i such that x ∈ A 1 and y ∈ A 2 . To ease the computational process we make use of the recently developed concept where these quotient graphs are reduced to a strenght-weighted graph with vertex and edge sets consisting of their corresponding strenght-weighted parameters.
A graph G with strength-weighted functions (SW V , SW E ) assigned to the vertex set V (G) and edge set E(G) is a strength-weighted graph [2] while SW E is the pair (w e , s e ) of an edge weight function w e : E(G) → R + 0 and a strength function s e : E(G) → R + 0 . The distance function of the strength-weighted graph G sw remains the same as in the graph G, while the degree and the bond closeness set parameters of a vertex u and an edge f = uv are defined as follows.
Hence, the Szeged-like indices of G sw can be in the form such that simple Szeged, PI and Mostar will take weighted measure as the edge strength value s e and wSz • The seminal paper [12] developed the cut method for the Wiener index and for the case when Θ is transitive. In [13], the method was extened to general graphs, that is, to compute the Wiener index of an arbitrary graph no matter whether Θ is transitive or not. Here we consider the currently most general set-up of the cut method in terms of strength weighted graphs as recently proposed in [2] as follows. Let G be a molecular graph with the Θ * -partition E(G) = {E 1 , . . . , E p }. Then and in particular,

Szeged-like indices of melem structures
There are two bridging bonds between two consecutive heptazines, we shall denote them F B i and SB i , 1 ≤ i ≤ s − 1. Due to chain arrangement, the graph theoretical measures of F B i are equivalent to those of SB s−i and hence, we restrict our computation to only F B i . As before, the quotient graph MC[s]/F B i has partite set {A b i } and We now reckon the bonds of heptazines based on the horizontal and slanting types. For 1 ≤ i ≤ s, let F H i and SH i be the Θ-classes constructed from the first and second layers horizontal bonds of i th heptazine. The quotient graphs produced by F H i and SH i are also K 1,1 , but the vertex weights, vertex strengths, edge weights, edge strength for F H i and SH i are 4, 15s − 3, 3, 18s − 5, 10, 12, 2 and 9, 15s − 8, 9, 18s − 12, 16, 21, 3 respectively. As we did for horizontal bonds, let F O i and SO i (1 ≤ i ≤ s) be the Θ-classes constructed from the first and second layers' obtuse bonds of i th heptazine. Then the weighted measures for the quotient graphs K 1,1 produced by F O i and SO i are respectively 15i − 11, 15(s − i) + 12, 18i − 15, 18(s − i) + 13, 10, 12, 2 and 15i − 6, 15(s − i) + 7, 18i − 9, 18(s − i) + 6, 16, 21, 3. Finally, let F A i and SA i (1 ≤ i ≤ s) be the Θ-classes constructed from the acute bonds and the graph theoretical quantities of the i th obtuse Θ-class are equivalent to (s−i+1) th acute Θ-class due to their symmetrical nature. Therefore, the computation of all forms of Szeged-like indices can be done from the following equation.

Theorem 3.2. Let MR[s] be a melem chain of dimension s ≥ 2.
Proof. We use two different cases to compute the Szeged-like indices of melem rings based on the odd and even number of melem units. In the case of even s, we identify the appropriate Θ * -classes and then use a strength-weighted quotient graph to obtain the necessary results. But if s is odd, the strength-weighted quotient graph is more complex, and hence we partition the bonds of corresponding quotient graph based on the cardinalities of the closeness sets of terminal vertices.   Here F H i and SH i generate an identical quotient graph K 1,1 , but the vertex weights, vertex strengths, edge weights, edge strength for F H i and SH i are 4, 15s − 4, 3, 18s − 5, 10, 12, 2 and 9, 15s − 9, 9, 18s − 12, 16, 21, 3 respectively. We compute the required indices by simplifying the following equation.

Case 2 (s odd):
In this case, we use the Θ-classes P P i , F H i and SH i , where 1 ≤ i ≤ s as in Case 1. It is important to note that all the slanting and bridging bonds belong to the single Θ * -class SB and the corresponding quotient graph is displayed in Figure 3. We now classify the bonds of the quotient graph by considering the cardinalities of closeness sets in which the graph theoretical quantities are given in Table 1. Hence, the Szeged-like indices are derived from the following equation.  Class |c i |