Degree-Based Topological Index (DBTI)

• DBTI is a graph invariant defined by a symmetric function over vertex degrees, capturing structural information in molecular graphs.
• It encompasses classical indices like Zagreb, Randić, and Sombor, and supports various generalizations including reverse and distance-augmented forms.
• DBTIs are vital in QSPR/QSAR modeling, enabling accurate prediction of physicochemical properties through efficient computational methods.

A degree-based topological index (DBTI) is a graph invariant central to chemical graph theory, defined for a molecular graph $G=(V,E)$ by applying a symmetric function of vertex degrees over all edges: $I(G)=\sum_{uv\in E} f(d(u),d(v))$. DBTIs encode information on connectivity, molecular branching, and vertex environments, and serve as key descriptors in quantitative structure–property/activity relationships (QSPR/QSAR) for predicting physicochemical and biological properties of molecules. The class of DBTIs encompasses classical indices such as the Zagreb, Randić, atom–bond connectivity, and Sombor indices, along with numerous generalizations, reverse forms, and distance-augmented variants.

1. Core Definitions and Classical Families

Let $G=(V,E)$ be a simple undirected graph, with $d(u)$ the degree of vertex $u$. A DBTI is any invariant of the form

$I(G) = \sum_{uv\in E}f(d(u),d(v)),$

where $f(x,y)$ is symmetric. Prominent subfamilies include:

• General Randić index: $R_\alpha(G) = \sum_{uv\in E}(d(u)d(v))^\alpha$, capturing multiplicative vertex degree effects; $\alpha=-\frac{1}{2}$ yields the original Randić index.
• First and second Zagreb indices: $M_1(G)=\sum_{uv\in E}[d(u)+d(v)]$, $I(G)=\sum_{uv\in E} f(d(u),d(v))$0, encoding linear and quadratic degree contributions.
• Atom–bond connectivity (ABC) index: $I(G)=\sum_{uv\in E} f(d(u),d(v))$1, empirically correlated with thermal properties.
• Geometric–arithmetic (GA) index: $I(G)=\sum_{uv\in E} f(d(u),d(v))$2.
• Sombor index: $I(G)=\sum_{uv\in E} f(d(u),d(v))$3, reflecting Euclidean combination of end-degrees.
• First Banhatti–Sombor index: $I(G)=\sum_{uv\in E} f(d(u),d(v))$4 (Lin et al., 2021).

Degree-based indices can also be generated via the M-polynomial $I(G)=\sum_{uv\in E} f(d(u),d(v))$5 (where $I(G)=\sum_{uv\in E} f(d(u),d(v))$6 counts edges of degree $I(G)=\sum_{uv\in E} f(d(u),d(v))$7), enabling routine generation of DBTIs by suitable differential operators (Deutsch et al., 2014).

2. Generalizations and Structural Parameterizations

DBTIs admit extensive structural and functional generalizations:

• (a, b)-Zagreb indices: $I(G)=\sum_{uv\in E} f(d(u),d(v))$8 generalize both classical Zagreb and Randić families (Sarkar et al., 2019).
• Reverse degree indices: Utilizing the "reverse degree" $I(G)=\sum_{uv\in E} f(d(u),d(v))$9, reverse DBTIs (e.g., $G=(V,E)$0) emphasize peripheral over highly connected vertices (Nagesh, 2024).
• Distance-augmented variants: $G=(V,E)$1-distance degree-based indices define for each $G=(V,E)$2 a $G=(V,E)$3-distance degree $G=(V,E)$4 (number of vertices at distance $G=(V,E)$5), leading to families such as the leap Zagreb and leap Sombor indices, especially relevant for extended benzenoid systems (Lal et al., 2022).
• Neighborhood degree indices: Functions of neighbor sum degrees (e.g., $G=(V,E)$6) have also been proposed for finer QSPR/QSAR modeling (Mondal et al., 2019).
• Bond Incident Degree (BID) indices: Any index of the form $G=(V,E)$7, with $G=(V,E)$8 symmetric and nonnegative, subsumes virtually all DBTIs of interest (Ali et al., 2017).

3. Extremal Structure and Optimization Techniques

The extremal analysis of DBTIs reveals deep connections between degree sequence structure and the index values:

• Majorization and Schur-convexity: For functions $G=(V,E)$9 convex or Schur-convex, extremal values are realized on graphs with maximally imbalanced (for maxima) or uniform (for minima) degree sequences. Rigorous bounds in $d(u)$0-cyclic graph families are obtained by evaluating the index on maximal/minimal sequences under majorization (Bianchi et al., 2013).
• Hub-degree Principle: For monotone or convex $d(u)$1, extremal $d(u)$2-graphs invariably possess a vertex of degree $d(u)$3 (the "hub"), with remaining edges distributed to maximize/minimize the target function (Ali et al., 2017).
• Polyhedral and Linear Programming Frameworks: In low-degree chemical graphs ($d(u)$4), all possible degree-pair distributions form a polytope in $d(u)$5; thus, extremal DBTI values correspond to extreme points, computable via linear optimization. This approach underlies the ChemicHull system for extremal structure discovery (Bonte et al., 25 Nov 2025).
• Asymptotics on Random Graphs: In sufficiently dense heterogeneous Erdős–Rényi random graphs, all DBTIs of the form $d(u)$6 are asymptotically normal with computable mean and variance; the general Randić index exhibits a phase transition in its fluctuation rate at $d(u)$7 (Yuan, 2023). In random chain models, DBTIs obey CLT and SLLN, with explicit moment formulas (Sigarreta et al., 2022).

4. Analytical and Closed-Form Evaluations on Structured Graph Families

The analysis of DBTIs for large regular structures leverages combinatorial counts:

• Edge-type partitioning: For lattices, planar octahedron networks, silicate/oxide/honeycomb/hexagonal graphs, and molecular graph templates, edge sets are partitioned by vertex degree pairs to yield closed-form DBTI expressions as polynomials in system size (Sarkar et al., 2019, Ali et al., 2019, Wang et al., 2016).
• Caterpillars and chemical trees: For trees with constrained number of leaves or maximum degree, strict convexity of $d(u)$8 determines the degree sequence attaining extremal $d(u)$9 (Hamoud et al., 6 Aug 2025).
• Benzenoid systems: $u$0-distance degree DBTIs discriminate sharply between zigzag and rhombic architectures and admit explicit polynomial formulas for all eight $u$1-distance indices (Lal et al., 2022).

5. Applications and Correlations in Chemoinformatics

• QSPR/QSAR utility: DBTIs, both classical and reverse, consistently afford strong linear correlations ($u$2) to boiling point, enthalpy of vaporization, molar volume/refraction, polar surface area, and polarity—demonstrated for hypomethylating agents and alkane isomers in multiple studies (Nagesh, 2024, Mondal et al., 2019). Reverse DBTIs, in particular, detect structural motifs (e.g., terminal functional groups) often decisive for intermolecular interactions.
• Algorithmic computation: Efficient $u$3 algorithms for DBTI calculation are available for molecular graphs, especially when leveraging the M-polynomial or neighbor-degree approaches (Deutsch et al., 2014, Mondal et al., 2019).
• Graph invariance: All DBTIs are isomorphism invariants by construction, permitting universal application as graph descriptors.
• Descriptive power variants: Neighborhood degree and R-degree–based indices extend classical topological descriptors by encoding higher-order local environments and clustering effects, increasing discrimination among isomers (Ediz, 2017).

6. Open Problems and Methodological Extensions

• Characterization of extremal graphs for arbitrary DBTIs in broad graph classes remains a major subject of research; recent polyhedral and majorization approaches offer systematic frameworks (Bonte et al., 25 Nov 2025, Bianchi et al., 2013).
• Asymptotic theory for DBTIs on random graphs and random molecular chains has revealed normal limit laws and described scaling regimes for index variance, including phase transitions (e.g., in the general Randić index) (Yuan, 2023, Sigarreta et al., 2022).
• Integration with QSPR modeling: The explanatory power of newly introduced indices and their variants (e.g., reverse and distance-based DBTIs) for experimental physico-chemical endpoints is an ongoing empirical and analytical challenge.
• Higher-dimensional structures: Extension of DBTI analysis to hypergraphs, directed graphs, and non-planar lattices is an active direction.

7. Summary Table of Principal DBTIs

Index Name	Formula over edges	Parameterization
First Zagreb	$u$4	additive
Second Zagreb	$u$5	multiplicative
Randić	$u$6	exponent $u$7
Sombor	$u$8	quadratic
Atom–bond connectivity	$u$9	fractional
Geometric–arithmetic	$I(G) = \sum_{uv\in E}f(d(u),d(v)),$0	geometric/arithmetic
First Banhatti–Sombor	$I(G) = \sum_{uv\in E}f(d(u),d(v)),$1	inverse square root

DBTIs, via their canonical dependence on local and higher-order vertex degree structure, unify much of the theory and practice of structural graph descriptors in chemistry, physics, and network science. Recent developments, notably reverse degree formulations and computational/global optimization approaches, have expanded descriptive and predictive capacity, grounding QSPR modeling in robust mathematical invariants and enabling efficient search for extremal molecular architectures (Nagesh, 2024, Bonte et al., 25 Nov 2025).