An infectious virus called dengue infects people through the bite of infected mosquito species called Aedes. Tropical and subtropical regions are particularly at risk for dengue fever, which poses a significant public health threat. The signs and symptoms of dengue include high fever, severe headache, joint and muscle pain, rash, mild bleeding, pain behind the eyes, nausea, vomiting and mild respiratory problems. In severe cases, the symptoms may worsen and the individual may experience intense abdominal pain, persistent vomiting, rapid breathing, lethargy, restlessness, bleeding from the nose or gums, blood in vomit or stools and even organ failure. It also lead to drop in platelet count, which increases the risk of bleeding. It is hard to diagnose dengue fever since the symptoms of dengue is similar to many viral infections. Hence it is advised to run laboratory tests, such as reverse transcription-polymerase chain reaction (RT-PCR) or serological tests to diagnose and differentiate dengue from other infections. Numerous drugs have been evaluated through in vitro, in vivo and clinical studies to identify potential antivirals for dengue. In addition to these efforts, Dengvaxia, the first licensed dengue vaccine, has been developed to provide partial protection against dengue virus, although it is not a therapeutic antiviral. Furthermore, investigational drugs like JNJ-A07 and JNJ-1802, that target the dengue virus non-structural protein 4B (NS4B), have shown promising results in pre-clinical and clinical studies, demonstrating potent pan-serotype activity and potential to inhibit viral replication effectively. Several drugs have shown potential antiviral activity against the dengue virus, including UV-4B (N-9-methoxynonyl-1-deoxynojirimycin), Lycorine, ST-148, 4-HPR, Silymarin, Baicalein, Quercetin, Naringenin, Nelfinavir, Ivermectin, Mosnodenvir (JNJ-1802), NITD-688, Metoclopramide, JNJ-A07, and Betulinic acid. The targets and mechanisms of these drugs in combating the dengue virus are summarized in Table 1.

Chemical graph theory (Wagner and Wang, 2018) is a part of graph theory which combines the principles of chemistry and graph theory. In chemical graph theory, the molecular structure of a chemical compound can be modelled in terms of an isomorphic molecular graph G ( V , E ) with atoms as vertex set V ( G ) and the bonds between the atoms as an edge set E ( G ) . The degree of an atom(vertex) v , denoted by d ( v ) , is the number of bonds(edges) incident to the atom v and the neighborhood of an atom v , denoted by N ( v ) , is the set of all atoms that are adjacent to v .

Topological index of a molecular structure is a numerical value computed based on the structure of a molecule graph. It converts the qualitative or abstract information of a molecule into a quantitative form. Various types of topological indices have been developed based on the different parameters of the molecular graph structures. These include distance-based indices, degree-based indices, neighborhood-based indices and connectivity-based indices. The Quantitative Structure-Property Relationship (QSPR) analysis of a molecular graph is carried out through the topological indices to establish mathematical relationships between the structural features of chemical compounds and their physical or chemical properties.

In recent studies, researchers have employed the Quantitative Structure-Property Relationship (QSPR) analysis using various topological indices to predict the physicochemical and ADMET (Absorption, Distribution, Metabolism, Excretion, and Toxicity) properties of diverse drug compounds. Among the various topological index variants, degree-based and neighborhood degree-based indices have been widely employed to evaluate their predictive capabilities for drug-like compounds. For instance, Tamilarasi and Balamurugan (2025) utilized these indices to predict the properties of antifungal drugs, while Arockiaraj et al. (2025) applied them to analyze compounds used in the treatment of lung cancer. Similarly, degree-based indices have been employed in the QSPR modeling of drugs targeting heart disease Kuriachan and Parthiban (2025); Hasani and Ghods (2024), blood cancer Zaman et al. (2024) and tuberculosis Abubakar et al. (2024). Their application extends to respiratory diseases as well, with studies exploring treatments for asthma Balasubramaniyan and Chidambaram (2023) and COVID-19 Ugasini Preetha et al. (2024); Das et al. (2023). It is noteworthy that these degree-based indices have been widely applied across various diseases, highlighting their utility and predictive power. In addition to degree-based indices, distance-based topological indices have also proven effective. For instance, Sardar and Hakami (2024) employed the distance-based indices to predict properties of drugs used in Alzheimer’s disease, while Huang et al. (2023) focused on anticancer agents. Recent literature also highlights the use of more specialized topological variants. Density-based indices were applied to study monkeypox-related drugs Kalaimathi and Balamurugan (2023), while reverse-sum Revan indices found use in analyzing antifiloviral drugs Tamilarasi and Balamurugan (2022). Thilsath parveen and Siddiqui (2024) explored domination distance-based indices and Shi et al. (2025) investigated temperature-based indices to model the properties of anticancer compounds.

In graph theory (Bondy and Murty, 2008), a graph coloring of the graph G ( V , E ) is an assignment of colors to the elements of the graph such as vertices or edges or both. The coloring of each element of the graph holds significance in its own distinct manner. In this article, the vertex coloring of a graph is considered. The vertex coloring c : V ( G ) → N , where N is the set of natural numbers, is said to be proper if no two adjacent vertices have the same color. Here, the set of natural numbers represents the set of colors. The minimum number of colors used to color the vertices of the graph is called as chromatic number and it is denoted as χ ( G ) . The graph coloring helps to study the structural properties of graphs by analysing the relationship between the number of colors used to color the graphs and various graph parameters such as vertex degree, connectivity, independent number, neighborhood set and more.

The graph coloring finds various applications in chemistry, particularly in representing molecular structures. The assignment of colors to the vertices helps in differentiating the types of atoms or functional groups within a molecule (Huckvale et al., 2023; Jin et al., 2020). In reaction network analysis, vertices are represented as chemicals and the two vertices are connected by an edge if the two chemicals are reactive with each other. Coloring these vertices facilitates the separation of reactive chemicals, aiding the chemical manufacturing industry in efficiently (optimally) storing non-reactive chemicals together in their warehouses. The minimum number of colors used determines the minimum number of compartments or rooms required for storing the chemicals. Graph coloring is also applicable in conformational analysis, enabling the identification of structural similarities or differences. Different isomers can be colored distinctly, contributing to the systematic exploration of atom alignments in a molecular structure.

The topological indices based on graph coloring can provide a comprehensive understanding of the molecular graphs, facilitating the prediction of the physical and chemical properties of the molecules. Therefore, the chromatic topological indices emerges in the field of chemical graph theory. Unlike traditional indices, these coloring-based indices provide an alternate method for analyzing molecular structures, to understand how the arrangement of colors influences the molecular properties.

The notion of chromatic topological indices was introduced by Johan kok et al. in (Kok et al., 2016). For any color set, C = { c 1 , c 2 , … , c l } and the coloring ϕ t ( G ) for 1 ≤ t ≤ l ! of a graph G , Johan kok et al. (Kok et al., 2016) introduced the indices viz., first chromatic Zagreb index M 1 ϕ t ( G ) , second chromatic Zagreb index M 2 ϕ t ( G ) and third chromatic Zagreb index M 3 ϕ t ( G ) . The arrangement of colors in a graph is categorized as ϕ − and ϕ + coloring. In ϕ − coloring, colors are assigned to the vertices in increasing order, maximizing the usage of each color before proceeding to the next color. In ϕ + coloring, colors are assigned to the vertices in decreasing order, maximizing the usage of each color before proceeding to the next color.

In (Albina and Manonmani (2022); (2021); (Kok et al. (2017); Rose and Naduvath (2018)) the chromatic topological indices of some classes of graphs were determined. Following this, Smitha) Rose and Sudev Naduvath introduced several variants of chromatic indices viz., chromatic total irregularity index (Rose and Naduvath, 2020a), injective chromatic zagreb indices (Rose and Naduvath, 2019), injective chromatic total irregularity index (Rose and Naduvath, 2019), equitable chromatic zagreb indices (Rose and Naduvath, 2020b) and equitable chromatic irregularity index (Rose and Naduvath, 2020b). In (Rose and Naduvath, 2019) and (Rose and Naduvath, 2020b), they computed the injective and equitabe chromatic Zageb indices and injective and equitabe chromatic total irregularity index for the Mycielskian graphs of path and cycle. Later, in (Rose and Naduvath, 2020a), they computed the chromatic total irregularity index for path and cycle.

Motivated by the exploration of various coloring-based topological indices, six new chromatic topological indices and ten new induced color-based topological indices are introduced in this article. The induced color-based indices distinguish themselves by providing a unique method to analyze the molecular structures through the color sum of the vertices.

The coloring-based topological indices have not yet been explored especially in the context of their effectiveness in predicting properties through QSPR analysis. To address this research gap, the performance of chromatic topological indices and induced color-based indices are investigated in this article in the context of molecular graph modeling for QSPR analysis. The coloring techniques considered, namely, the proper vertex coloring and sigma coloring have distinct theoretical significance in structural analysis of the molecular structures. The relevance of these coloring techniques and their foundational importance are discussed in detail in the Section 3.1.

Specifically, the induced color-based and chromatic topological indices are computed for 15 potential antivirals of dengue disease. QSPR analysis is performed through these indices and linear regression to explore the physicochemical properties of dengue antivirals. Further, the comparative analysis of the two types of indices is performed to identify the potential indices to predict the properties of drugs.

The concept of isomorphic molecular graph of a chemical structure is discussed in this section.

In this section, the concept of vertex coloring relevant to this article is discussed. Furthermore, the concept of sigma coloring and its significance are presented with appropriate examples.

Definition 2:

Let G ( V , E ) be the isomorphic molecular graph. The vertex coloring c of G is said to be proper vertex coloring if no two adjacent vertices are assigned with the same color.

Definition 3:

Let G ( V , E ) be a graph with vertex set V and edge set E . Let C = { c 1 , c 2 , … , c ℓ } be the set of colors used to color the vertices of G . A ϕ − coloring is a vertex coloring in which the colors in C are assigned in an order starting from the first color c 1 . The color c 1 is assigned to the maximum possible number of vertices, followed by the color c 2 is assigned to the maximum possible number of the remaining uncolored vertices. This process is continued sequentially with c 3 , c 4 , … , c ℓ until all vertices in V are colored.

Definition 4:

Let G ( V , E ) be a graph with vertex set V and edge set E . Let C = { c 1 , c 2 , … , c ℓ } be the set of colors used to color the vertices of G . A ϕ + coloring is a vertex coloring in which the colors in C are assigned in order starting from the last color c ℓ . The color c ℓ is assigned to the maximum possible number of vertices, followed by the color c ℓ − 1 is assigned to the maximum possible number of remaining uncolored vertices. This process is continued sequentially with c ℓ − 2 , c ℓ − 3 , … , c 1 until all vertices in V are colored.

Definition 5:

Let G ( V , E ) be a graph with vertex set(atom set) V ( G ) and edge set (bond set) E ( G ) . The neighborhood of a vertex(atom) v is defined as the set of vertices(atoms) that are adjacent to v . The cardinality of N ( v ) denoted as | N ( v ) | is defined as the number of vertices(atoms) in N ( v ) .

Definition 6:

(Chartrand et al., 2010) Let G ( V , E ) be a isomorphic molecular graph, where V ( G ) represents the vertex set(atom set) and E ( G ) represents the edge set(bond set). Let c : V ( G ) → N be a vertex(atom) coloring function, where N is a set of natural numbers. The color sum of the atom v ∈ V ( G ) , denoted by σ ( v ) is the sum of colors of the neighbor atoms of v . The coloring c is called sigma coloring if σ ( u ) ≠ σ ( v ) for any two atoms(vertices) u and v that shares a common bond(edge). The minimum number of colors needed for sigma coloring is called as sigma chromatic number of G and it is denoted by σ ( G ) .

Throughout the article, the values inside the circles represent the proper vertex coloring, while the values outside the circles correspond to the sigma coloring. In sigma coloring, the values outside the brackets is the initial color assigned to the vertex and the values inside the bracket denote the color sum of the vertex.

Example 1:

The chemical structure of dopamine is shown in Figure 2a and the vertex coloring and sigma coloring of the isomorphic molecular graph of Dopamine are shown in Figure 2b.

Let G ( V , E ) be an isomorphic molecular graph, where V represents the atom set and E represents the set of bonds. The ten new induced color-based topological indices and six new chromatic topological indices are introduced in this article and are defined in Table 2.

The induced color-based topological indices are computed using the graph coloring variants, where the vertex colors are derived from an initial assignment of vertex or edge colors. Examples of some of such variants include sigma coloring, closed sigma coloring, additive coloring, modular coloring, closed modular coloring, antimagic labeling and lucky labeling. In induced color based topological indices, α ( u ) represents the induced vertex color. The chromatic topological indices are determined using the proper vertex coloring technique, where c ( u ) represents the color assigned to the vertex u.

The induced color-based topological indices capture the influence of neighboring vertices or incident edges, whereas chromatic indices utilize graph coloring attributes to represent molecular features.

Let C = { c 1 , c 2 , … , c l } be the set of colors used to color the vertices of graph that satisfies the condition of the employed coloring. Let each vertex u ∈ V ( G ) is assigned a color c ( u ) = c j for 1 ≤ j ≤ l . Let ϕ t : C → { 1,2 , … , l } denote the t t h permutation of numbers { 1,2 , … , l } , for 1 ≤ t ≤ l ! . The value of t ranges from 1 to l ! , since ϕ is a permutation function using l colors,. Then, for each such ϕ t , the novel induced color-based topological indices and chromatic topological indices are defined in Table 2.

Let C = { c 1 , c 2 , … , c l } be a proper vertex coloring of a graph G , where each vertex u ∈ V ( G ) is assigned a color c ( u ) = c j for 1 ≤ j ≤ l . Let ϕ t : C → { 1,2 , … , l } denote the t t h permutation of numbers { 1,2 , … , l } , for 1 ≤ t ≤ l ! . The value of t ranges from 1 to l ! , since ϕ is a permutation function using l colors. Then, for each such ϕ t , the chromatic topological indices introduced by Kok et al. (2016) and Rose and Naduvath (2020a) are defined as follows:

1. The first chromatic Zagreb index of G is defined as, M 1 ϕ t = ∑ j = 1 l θ ( c j ) j 2 , where θ ( c j ) is the number of vertices colored with the color c j in G .

Let ϕ − denote the permutation ϕ t that yields the minimum value of the topological index among all l! possible permutations and let ϕ + refers to the permutation ϕ t that gives the maximum value. Let InV be the topological index value correspond to the given permutation. Then, the relationship between the topological index value is given by I n V ϕ − ≤ I n V ϕ t ≤ I n V ϕ +

These topological indices form the basis for QSPR analysis to improve the predictive accuracy of analysis in determining the physicochemical properties of antiviral drugs of dengue. The induced color-based topological indices are computed using a graph coloring variant known as sigma coloring, while the chromatic topological indices are calculated using the proper vertex coloring.

A systematic approach is employed to analyze the properties of potential antiviral drugs for dengue disease through the Quantitative Structure-Property Relationship (QSPR) graph modelling by considering the topological descriptors of the isomorphic molecular graphs of the drugs. The methodology consists of the following key steps:

1. Data Collection and Isomorphic Molecular Graph Construction

The most effective indices were identified based on their predictive power and statistical significance, offering a robust framework for understanding the molecular properties of antiviral drugs used in dengue treatment.

In this section, the ϕ − and ϕ + chromatic and induced color-based topological indices of potential antivirals of dengue disease are computed through proper vertex coloring and sigma coloring respectively. The two coloring scheme ϕ − and ϕ + are employed to investigate how the order of color assignment influence the predictive power of the topological indices. The chemical structure of the considered potential antivirals of the dengue disease are obtained from the National Center for Biotechnology Information (NCBI) whose URL is pubchem.ncbi.nlm.nih.gov and they are provided in the Supplementary Material.