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Atom-Bond Connectivity Index

A degree-based graph descriptor that sums √[(d(u)+d(v)−2)/(d(u)d(v))] once over every edge to produce a label-independent scalar.

Version
v1 · 2026-09-28 · History
Domain-specific #
8060
Domain group
Natural Sciences
Origin domain
Chemistry & Materials Science
Subdomains
Chemical Graph Theory, Molecular Topological Indices → Chemistry & Materials Science
Aliases
ABC index, Atom–bond connectivity index

Core Idea

The atom–bond connectivity index (ABC) is a degree-based graph descriptor defined by ABC(G)=Σ_{uv∈E(G)}√[(d(u)+d(v)−2)/(d(u)d(v))]. Each edge contributes a weight determined by its endpoints' degrees; summing all edges gives one scalar unchanged by vertex relabeling.

ABC was introduced as a molecular structure descriptor and has been used in scoped property models. The graph value is exact once representation is fixed, while chemical prediction is empirical and dataset-dependent. Changing the edge formula or degree input creates another index.

Nonisomorphic graphs can share the same scalar, so ABC is compact rather than structurally complete.

Its usefulness ultimately depends on whether that compression preserves task-relevant differences.

How would you explain it like I'm…

Stick-Score Molecule Number

Think of a molecule as balls joined by sticks. For each stick, look at how many sticks each of its two balls has, and use those two counts to give the stick a little score. Add up every stick's score and you get one number for the whole molecule, called the atom-bond connectivity index.

Molecule Connection Score

Chemists sometimes draw a molecule as a graph: dots for atoms and lines for bonds. The Atom-Bond Connectivity Index turns that whole drawing into one number. For every line, you look at how many lines touch each of its two end dots, which is called their degree. A formula turns those two degrees into a small score for that line, and you add up the scores for all the lines. The number doesn't depend on how you label the dots, and chemists have tried using it to help predict things about molecules, though how well that works depends on the molecules and the property.

Degree-Based Edge-Sum Index

The Atom-Bond Connectivity (ABC) index is a topological index: a single number computed from a graph's structure, often a molecular graph with atoms as vertices and bonds as edges. For each edge uv, take the degrees d(u) and d(v), the number of edges meeting each endpoint, and compute the square root of (d(u) + d(v) − 2) divided by d(u)·d(v). Summing this value over every edge once gives ABC(G). Because it depends only on the graph's structure, it is the same however the vertices are labeled. It was introduced in 1998 as a molecular descriptor and was related empirically to heats of formation for a set of alkanes. The index itself is exact once the graph is fixed, but its usefulness for predicting chemistry depends on how the molecule is represented, which data are used and which property is modeled. It is a different quantity from the Randić connectivity index and from similarly named variants.

 

The atom–bond connectivity (ABC) index is a degree-based topological index defined for a graph G as ABC(G) = Σ over edges uv of √[(d(u) + d(v) − 2)/(d(u)d(v))], where d(u) and d(v) are the endpoint degrees and each edge is counted once. Each edge thus receives a weight determined solely by its local degree pair, and summation yields a label-independent scalar invariant of the whole graph. Ernesto Estrada, Luis Torres, Lissette Rodríguez and Ivan Gutman introduced it in 1998 as a molecular graph descriptor, relating it empirically to heats of formation for a studied set of alkanes. Two senses of exactness must be kept apart: the index is an exact graph invariant once the carrier graph is fixed, whereas its chemical predictive meaning depends on the molecular representation, dataset, target property and fitted model. ABC is distinct from the Randić connectivity index and from variants with similar names: the atom–bond sum-connectivity index changes the denominator of the edge term, the Graovac–Ghorbani index uses distance-based vertex counts, and exponential versions transform the local contribution. Those variants can be useful descriptors but are not the standard ABC index.

Scope of Application

It applies where a finite graph and degree convention are explicit enough for the standard edge sum.

  • Chemical graph theory — Molecular graphs supply atoms, bonds, and endpoint degrees.
  • Topological descriptors — ABC compresses local connectivity into one graph-level scalar.
  • Extremal graph theory — Bounds and optimizing structures are studied for graph classes.
  • QSPR modeling — Validated models relate ABC to defined properties on stated domains.
  • Descriptor comparison — Analysts test discrimination and complementarity with other indices.
  • Algorithmic enumeration — Computation screens many candidate graphs under one formula.

Hydrogen convention, bond representation, dataset, and property model must be declared; one correlation is not a universal guarantee.

Clarity

Atom–Bond Connectivity Index separates molecular representation, exact invariant calculation, and empirical interpretation. It is not an experimental property measurement or complete molecular encoding. The formula provides a decisive identity test: using distance counts, a degree-sum denominator, an exponential transform, or another edge weight yields a related descriptor, not the standard ABC index.

Manages Complexity

A graph can contain many vertices, bonds, branches, cycles, and local environments. ABC collapses endpoint-degree pairs into a fixed local-to-global sum, making comparison and model input compact. The compression loses geometry, labels, stereochemistry, electronic structure, and some graph discrimination, so underlying graph, formula version, degree-pair distribution, and model domain remain essential provenance.

Abstract Reasoning

Use representation control, edge-local computation, and validation separation. Define the graph, compute every vertex degree, apply the exact weight to each edge once, and verify relabeling invariance. For mathematical work, connect degree constraints to bounds; for QSPR work, fit and validate separately. Ask whether nonisomorphic structures collide and whether another descriptor supplies missing information.

Knowledge Transfer

The computation transfers across finite graphs when carrier and degree conventions are explicit. Chemical interpretation transfers only across matched molecular representations and validated property domains. ABC is a strict child of Measurement because a fixed procedure maps graph structure to a scalar descriptor; that parent does not imply direct instrumentation or guaranteed prediction. Ratio is not the parent: the output is an aggregate of transformed edge terms.

Relationships to Other Abstractions

Local relationship map for Atom-Bond Connectivity IndexParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Atom-BondConnectivity IndexDOMAINPrime abstraction: Measurement — is a kind ofMeasurementPRIME

Current abstraction Atom-Bond Connectivity Index Domain-specific

Parents (1) — more general patterns this builds on

  • Atom-Bond Connectivity Index is a kind of Measurement Prime

    Atom-Bond Connectivity Index is a strict kind of Measurement: A degree-based graph descriptor that sums √[(d(u)+d(v)−2)/(d(u)d(v))] once over every edge to produce a label-independent scalar.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Atom-Bond Connectivity Index sits in a sparse region of the domain-specific corpus (88th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08