Degree-Based Topological Indices

Updated 11 December 2025

Degree-based topological indices are graph invariants expressed as sums or products over edges/vertices using functions of incident vertex degrees, playing a key role in chemical graph theory.
They are computed via algebraic tools like the M-polynomial and operator calculus, which facilitate analysis of extremal structures and optimization over graph classes.
Applications include QSAR/QSPR modeling and network analysis, with indices such as the Zagreb, Randić, and Sombor indices used to predict physicochemical properties.

A degree-based topological index is a graph invariant defined as a sum (or, in special cases, a product) over edges or vertices, where the contribution from each edge or vertex depends solely on the degrees of the incident vertices. These indices have central importance in mathematical chemistry and network theory, serving as molecular descriptors and predictors of physicochemical properties.

1. Formal Definition and Families

Given a simple graph $G=(V,E)$ with degree function $d(v)$ , a degree-based topological index (DBTI) is any invariant of the form

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

where $f:\mathbb{N}\times\mathbb{N} \to \mathbb{R}$ is a symmetric function (i.e., $f(x,y) = f(y,x)$ ). Alternatively, vertex-based forms such as $I(G)=\sum_{v\in V} h(d(v))$ are common, especially for multiplicative and general Zagreb indices. Degree-based indices include:

First Zagreb index: $M_1(G)=\sum_{v\in V} d(v)^2 = \sum_{uv\in E}(d(u)+d(v))$
Second Zagreb index: $M_2(G)=\sum_{uv\in E} d(u)d(v)$
General Randić index: $R_\alpha(G)=\sum_{uv\in E}[d(u)d(v)]^\alpha$
Sombor index: $SO(G)=\sum_{uv\in E}\sqrt{d(u)^2+d(v)^2}$
Forgotten index: $F(G)=\sum_{v\in V}d(v)^3$
Bond incident degree (BID) indices: $BID(G) = \sum_{uv\in E}\Psi(d(u),d(v))$ , for non-negative symmetric $\Psi$
Multiplicative indices: $\prod_{uv\in E} F_E(d(u),d(v))$ or $\prod_{v\in V} F_V(d(v))$

Weighted and generalizations (e.g., hyper-Zagreb, leap-indices, neighborhood-degree versions) are also treatable within this framework (Deutsch et al., 2014, Yuan, 2023, Lal et al., 2022).

2. Algebraic and Combinatorial Frameworks

The M-polynomial, introduced by Deutsch and Klavžar, encodes the full edge-degree distribution in a graph: $M(G;x,y) = \sum_{1\leq i\leq j} m_{ij}(G)x^iy^j$ where $m_{ij}$ counts the edges connecting degrees $i$ and $j$ . Any degree-based index of the form $I(G) = \sum_{uv\in E} f(d(u),d(v))$ can be expressed as

$I(G) = \sum_{i\leq j} m_{ij}(G)f(i,j)$

and, for polynomial (or suitably smooth) $f$ , can be extracted via operator calculus (using $D_x=x\frac{\partial}{\partial x}$ , etc.) acting on the M-polynomial (Deutsch et al., 2014, Mehiri, 16 Nov 2025).

The polyhedral method, crucial for chemical graphs with bounded maximum degree (Δ ≤ 3), reduces optimization of DBTIs over chemical graph classes to linear programming over the polytope of feasible edge-type counts (Dusollier et al., 24 Jun 2025, Bonte et al., 25 Nov 2025).

3. Extremal Structure and Bounds

Extremal (maximizing/minimizing) graphs for DBTIs, notably in trees, chemical trees, unicyclic, and c-cyclic graphs, have been fully characterized for many indices:

For trees with given number $n$ $n$ of pendant vertices, $M_1$ $M_{1}$ attains its minimum on "4-trees" (all internal vertices of degree 4 if $n$ $n$ even, one degree-3 otherwise), $M_1(T)\geq 9n-16$ $M_{1} (T) \geq 9 n - 16$ . Conversely, $M_2$ $M_{2}$ is minimized by "stem-and-3-tree" constructions for $n\geq9$ $n \geq 9$ :
- Internal nodes of degree 3, with stems of degree 4 or 5 appending the required number of pendants (Goubko et al., 2014).
In BID-index extremality, graphs maximizing monotone BID-type indices always contain a universal vertex (degree $n-1$ ), typically realizing the star or star-plus-minor-perturbations structure (Ali et al., 2017, Hamoud et al., 6 Aug 2025).
For c-cyclic graphs ( $0\leq c\leq 6$ ), majorization of the degree sequence gives sharp Schur-convex/concave bounds for general Zagreb and multiplicative Zagreb indices, with explicit extremal degree sequences constructed via degree-sum inequalities (Bianchi et al., 2013).
If a forbidden subgraph is imposed (clique, cycle, bipartite graph), the maximum of a DBTI is achieved on the Turán-type or Fūredi extremal graph, depending on index monotonicity and subgraph structure (Gerbner, 2024).

Table: Selected sharp lower bounds and extremal structures in trees

Index	Lower Bound	Extremal Structure
$M_1$	$9n-16$, $n$ even	all internal deg 4 (4-tree)
$M_1$	$9n-15$, $n$ odd	1 internal deg 3, others 4
$M_2$	$n^2$ , $2\leq n\leq 8$	star $K_{1,n}$
$M_2$	$11n-27$, $n \geq 9$	stem-and-3-tree

(Goubko et al., 2014)

4. Asymptotic and Probabilistic Results

In random graph models, degree-based indices admit precise asymptotic laws:

For the heterogeneous Erdős–Rényi model, normalized DBTIs satisfy a central limit theorem with explicit mean and variance determined by the first and second degree-moments and derivatives of $f$ . In the general Randić family, the fluctuation regime undergoes a phase transition at τ=–½: for τ>–½, variance grows as $n(n\rho)^{4\tau+1}$ , while for τ=–½, variance is $\Theta(n)$ (Yuan, 2023).
On random chains built from standard molecular fragments, DBTIs grow linearly in chain length, with deterministic behavior for indices where the edge-contribution increment does not depend on attachment type (Sigarreta et al., 2022).
For large dense random networks, normalized logarithms of vertex-based or edge-based multiplicative indices ( $\langle\ln X_\Pi(G)\rangle/N$ ) scale with average degree, with explicit scaling functions for each index class (Aguilar-Sanchez et al., 2023).

5. Generalizations: Higher-Distance and Neighborhood-based Indices

Extended DBTIs incorporate information from beyond immediate adjacency:

$k$ -distance indices use, e.g., the 2-degree ( $\deg_2(v)$ ), i.e., the number of vertices at distance-2 from $v$ . Leap Zagreb, leap hyper-Zagreb, leap Sombor, and leap $Y$ indices are examples, and exhibit monotonic growth with system size and reflect medium-range topology in banded benzenoid fragments (Lal et al., 2022).
Neighborhood degree sum-based indices, e.g., $\Omega_G(v) = \sum_{u\in N_G(v)} d_G(u)$ , give rise to neighborhood forgotten, neighborhood second Zagreb, and related indices, which experimentally enhance discrimination and QSPR performance among chemical isomers (Mondal et al., 2019, Mondal et al., 2019).
The R-degree concept, $r(v) = S_v+M_v$ , with $S_v$ the sum and $M_v$ the product of neighboring degrees, yields R-indices that encode higher-order local environments (Ediz, 2017).

6. Multiplicative Formulations and Their Properties

Multiplicative degree-based indices capture nonlinear relationships and often grow super-exponentially with molecular size:

First multiplicative Zagreb: $\prod_{uv\in E}(d(u)+d(v))$
Second multiplicative Zagreb: $\prod_{uv\in E}(d(u)d(v))$ or $\prod_{v\in V} d(v)^{d(v)}$
Narumi-Katayama index: $\prod_v d(v)$

Explicit formulas are known for benzenoid systems, polycyclic hydrocarbons, and canonical lattice graphs. Analytical inequalities (Jensen-type, Kober, and Petrović bounds) relate additive and multiplicative forms (Kulli et al., 2017, Aguilar-Sanchez et al., 2023, Wang et al., 2016).

Table: Multiplicative indices in random graphs, dense limit

Index $X_\Pi$	$\ln X_\Pi / N$	Dense-Erdős–Rényi scaling
Narumi–Katayama	$\ln \langle k \rangle$	$\langle k \rangle$ = avg degree
First multiplicative Zagreb	$2 \ln \langle k \rangle$
Second multiplicative Zagreb	$\langle k \rangle \ln \langle k \rangle$

(Aguilar-Sanchez et al., 2023)

7. Applications and Chemical Significance

Degree-based indices underlie Quantitative Structure–Activity/Property Relationships (QSAR/QSPR), material informatics, and network property prediction:

Classical and advanced DBTIs correlate with boiling point, entropy, enthalpy of vaporization, acentric factor, and melting point in molecular datasets.
Neighborhood- and $k$ -distance-based indices demonstrate enhanced discriminating power for structural isomers in hydrocarbons and are more robust QSPR predictors (Mondal et al., 2019, Lal et al., 2022).
Extremal indices, under chemically relevant constraints (e.g., forbidden subgraphs, maximum degree), drive the design of molecular libraries and elucidate structure-activity bounds (Gerbner, 2024, Bonte et al., 25 Nov 2025).

The polyhedral/extremal approach, operationalized in the ChemicHull software, enables rapid identification of all potential extremal structures within prescribed graph classes, and helps resolve prior errors in extremal characterizations for certain DBTIs (Bonte et al., 25 Nov 2025).

References:

"Minimizing Degree-based Topological Indices for Trees with Given Number of Pendent Vertices + Erratum" (Goubko et al., 2014)
"M-Polynomial and Degree-Based Topological Indices" (Deutsch et al., 2014)
"Asymptotic distribution of degree--based topological indices" (Yuan, 2023)
"On the Extremal Graphs with Respect to Bond Incident Degree Indices" (Ali et al., 2017)
"On k-distance degree based topological indices of benzenoid systems" (Lal et al., 2022)
"Degree Based Topological Indices of a General Random Chain" (Sigarreta et al., 2022)
"New bounds of degree-based topological indices for some classes of $c$ -cyclic graphs" (Bianchi et al., 2013)
"Multiplicative topological indices: Analytical properties and application to random networks" (Aguilar-Sanchez et al., 2023)
"Generalized Multiplicative Indices of Polycyclic Aromatic Hydrocarbons and Benzeniod Systems" (Kulli et al., 2017)
"Bounds of Trees with Degree Sequence-Based Topological Indices on Specialized Graph Classes" (Hamoud et al., 6 Aug 2025)
"ChemicHull: an online tool for determining extremal chemical graphs of maximum degree at most 3 for any degree-based topological indices" (Bonte et al., 25 Nov 2025)
"On some new neighbourhood degree based indices" (Mondal et al., 2019)
"QSPR analysis of some novel neighborhood degree based topological descriptors" (Mondal et al., 2019)
"The general Zagreb index of lattice networks" (Sarkar et al., 2019)
"On R Degrees of Vertices and R Indices of Graphs" (Ediz, 2017)
"On extremal values of some degree-based topological indices with a forbidden or a prescribed subgraph" (Gerbner, 2024)
"Degree based Topological indices of Hanoi Graph" (Khalid et al., 2018)
"Complete polyhedral description of chemical graphs of maximum degree at most 3" (Dusollier et al., 24 Jun 2025)
"Further results on degree based topological indices of certain chemical networks" (Wang et al., 2016)
"Explicit M-Polynomial and Degree-Based Topological Indices of Generalized Hanoi Graphs" (Mehiri, 16 Nov 2025)