Comparison of the Inverse Degree Index for Certain Transformation Graphs

Rauf Irshad; Muhammad Hussain; Maqsood Ahmad

doi:10.22342/jims.v32i2.2074

Rauf Irshad ⁽¹⁾ , Muhammad Hussain ⁽²⁾ , Maqsood Ahmad ⁽³⁾

(1) Department of Mathematics, COMSATS University Islamabad, Pakistan,

(2) Department of Mathematics, COMSATS University Islamabad, Pakistan,

(3) Department of Mathematics, COMSATS University Islamabad, Pakistan

https://doi.org/10.22342/jims.v32i2.2074

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
May 29, 2025

Accepted
January 26, 2026

Published
June 17, 2026

Keywords:

inverse degree index, extremal graph, graph invariants, transformed graphs, complete graph

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Abstract

For a molecular graph $\Lambda,$ the inverse degree index (IDI) is defined by taking the sum over all vertices of the reciprocal degree of each vertex. The IDI is a structurally sensitive topological invariant that emphasizes low-degree vertices, making it effective for analyzing branching, irregularity, and deviations in molecular structures. Graph transformations prove to be an effective tool to construct new chemically meaningful structures. In this work, we study the behavior of IDI under four newly defined graph transformations. We first introduce families of $n$-vertex networks based on complete graphs of order $l \ge 3$, derive transformed networks, and establish inequalities involving the IDI. We then present extremal results for the transformed graphs and support our findings with computations on several representative examples.

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Authors

Rauf Irshad

Department of Mathematics, COMSATS University Islamabad

rauf.irshad118@gmail.com (Primary Contact)

Muhammad Hussain

Department of Mathematics, COMSATS University Islamabad

Maqsood Ahmad

Department of Mathematics, COMSATS University Islamabad

Irshad, R., Hussain, M., & Ahmad, M. (2026). Comparison of the Inverse Degree Index for Certain Transformation Graphs. Journal of the Indonesian Mathematical Society, 32(2), 2074. https://doi.org/10.22342/jims.v32i2.2074

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1. INTRODUCTION

Graph theory explores the interconnection network designs ranging from social and computer networks to molecular structures and contributes an inclusive understanding of such corresponding matters through their precise structure. The topology of any graph delivers knowledge regarding the techniques through which vertices are attached to the graph. In molecular graphs, vertices are considered to be atoms, and covalent bonds (double or triple) are taken as edges. Graph theory is an efective tool used in various research areas such as chemical graph theory (CGT), mathematical chemistry, and even coding theory [1,2,3]. Graph theory is the analysis of associations between various objects. Diferent graph notations are used to represent graphs, a graph is defined as a combination of a vertex and edge set $V ( \Lambda )$ and $E ( \Lambda )$ , respectively. There are various types of graphs, such as path, cycle, bipartite, complete, star, wheel graph, and many others. In this study, we work with simple connected graphs that are without multiple, directed, or weighted edges, and self-loops. In this paper, we are working with families of complete, star, cycle, and path graphs. A complete graph is a k-regular graph, where every vertex is connected to all other vertices. A star graph is a class of graphs in which only one vertex has degree $n - 1$ and $n - 1$ vertices have degree one. A cycle graph is denoted as $C _ { n } \ ( n \geq 3 )$ and every vertex has degree 2. Path graph $P _ { n }$ having exactly two vertices of degree one and all the others having degree 2.

Transformation graphs, akin to graph operations, help us in the development of new, complex, and important structures of our own choice. Transformation graphs hold the complete knowledge from the actual graph into a new transformed system. Therefore, if it is feasible to discover the provided graph from the transformed graph, then such a process may be utilized to figure out the structural properties of the initial graph, considering the transformation graphs for details, see[4]. In distinct mathematical literature, several graph transformations have been evaluated where the vertex set of the transformed graph is equal to $V ( \Lambda ) \cup E ( \Lambda )$ The most prominent classification of such graphs is the total graph. The total graph $T ( \Lambda )$ of a graph Λ is the graph whose vertex set is $V ( \Lambda ) \cup E ( \Lambda )$ and any two vertices of $T ( \Lambda )$ are adjacent if and only if they are either incident or adjacent in Λ[5].

A topological index (TI) transforms a molecular graph into a real number by involving diferent graph parameters like degrees, distances, and eigenvalues. Among these parameters, the connectivity or degree-related TIs are extensively investigated due to their close correlation with certain physicochemical properties of chemical species. Zagreb indices and their transformations are useful to anticipate about π-elctron energy of alternant hydrocarbons. TIs in certain combinations take part in the design of new drugs by participating in the analysis of quantitative activity and structure-property relationships. The analysis of TIs becomes very interesting for various research areas in the present era, particularly TIs impact on chemistry amongst the theoretical molecular descriptors, due to the prediction capability about physicochemical characteristics of various chemical substances. Some essential TIs of distinct networks were formerly computed in [6,7,8]. A chemist, Harold Wiener, initiated the idea of the pioneer distance-related index while working on boiling points of parafins and named it after his name, the Wiener index[9]. Gutman and Trinajesti´c [10,11], were the pioneers who provided the idea of degree-related indices while studying pi-electron energy of saturated hydrocarbons, and defined as follows:

$M_{1}(\Lambda)=\sum_{w\in V(\Lambda)}d_{\Lambda}^{2}(w)=\sum_{vw\in E(\Lambda)}[d_{\Lambda}(v)+d_{\Lambda}(w)]$

$M_{2}(\Lambda)=\sum_{vw\in E(\Lambda)}d_{\Lambda}(v)\cdot d_{\Lambda}(w)$

Later on, the analysis of Zagreb indices discovered many applications in QSPR and QSAR investigations [12,13]. Chemical applications and mathematical efects of Zagreb indices can be analyzed from [14,15,16]. The Zagreb indices have been manipulated to explore molecular complexity, chirality, ZE-isomorphism and hetero-systems. The specific modification of the Zagreb indices is defined with a variable parameter as, $\alpha \in \mathbb { R }$

These conceptions of the Zagreb indices appear to be preferably evaluated by Li et al[17,18]. There are some unique cases α, if $\alpha = 2$ then it is the first Zagreb index. For $\alpha = - 1$ is the inverse degree index (IDI) described as,

The name of this index was first introduced in (2005) and is also known as the modified total adjacency index[19]. For $\alpha = { \frac { - 1 } { 2 } }$ we obtained $( ^ { 0 } R _ { \frac { - 1 } { 2 } } ( \Lambda ) )$ zeroth-order connectivity R.I defined as,

For $\alpha = 3$ then,

this topological index is called the F index or forgotten index.

The IDI initially appeared in the software Grafiti [20,21] that generates conjectures and proved to be a promising predictor for various compounds in CGT, such as in discovering the physicochemical characteristics of various kinds of compounds like octane isomers, polychlorobiphenyls, and polyaromatic hydrocarbons [22,

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Comparison of the Inverse Degree Index for Certain Transformation Graphs

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