Wiener index¶

The sum of shortest-path distances over all unordered vertex pairs of a connected graph.

Version: v1 · 2026-09-08 · History
Domain-specific #: 7484
Origin domain: chemical graph theory
Subdomain: chemical graph theory

Core Idea¶

Introduced as a molecular topological descriptor, the Wiener index depends only on graph distances, admits weighted and disconnected variants under new conventions and appears in network and combinatorial analysis. All-pairs shortest-path lengths are computed in the molecular or abstract graph and aggregated once per unordered pair into a single global distance descriptor. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Wiener index belongs to chemical graph theory and is useful where the analyst can specify the typed chemical graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the graph and connectivity, vertex and edge representation, distance metric, unordered-pair convention, weighting and treatment of disconnected pairs are explicit. The scope is broad within that domain but bounded by the need for the graph and connectivity, vertex and edge representation, distance metric, unordered-pair convention, weighting and treatment of disconnected pairs are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the graph and connectivity, vertex and edge representation, distance metric, unordered-pair convention, weighting and treatment of disconnected pairs are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Wiener index can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Wiener index. Wiener index compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed chemical graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the graph and connectivity, vertex and edge representation, distance metric, unordered-pair convention, weighting and treatment of disconnected pairs are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of chemical graph theory because they reuse the typed chemical graph theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, All-pairs shortest-path lengths are computed in the molecular or abstract graph and aggregated once per unordered pair into a single global distance descriptor., and type the carrier, state every parameter and convention in the definition, test that the graph and connectivity, vertex and edge representation, distance metric, unordered-pair convention, weighting and treatment of disconnected pairs are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Wiener index Domain-specific

Parents (1) — more general patterns this builds on

Wiener index is a kind of Metric Prime

The proposed strict upward parent is prime:metric.

Hierarchy path (1) — routes to 1 parentless root

Wiener index → Metric → Function (Mapping)

Neighborhood in Abstraction Space¶

Wiener index sits in a crowded region of the domain-specific corpus (18^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Graph Invariants & Constructions (49 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08