Friendly-index set¶
Collect every edge-label imbalance attainable from nearly balanced binary vertex labelings of a graph, yielding a set-valued graph invariant rather than one selected labeling.
Core Idea¶
The friendly-index set of a graph is the set of absolute edge-label count differences attained by all friendly binary vertex labelings, where a labeling is friendly when its two vertex-label counts differ by at most one and edge labels are induced from endpoint labels modulo two. A balanced vertex labeling induces equal-endpoint and unequal-endpoint edge classes; their count imbalance is computed, and the attainable imbalances are unioned over all friendly labelings. 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¶
Friendly-index set belongs to graph theory and graph labeling and is useful where the analyst can specify a finite undirected graph with vertices, edges, and binary labels, then evaluate graph isomorphism preserves the set of attainable edge imbalances because it transports every friendly labeling and its induced edge counts. The scope is broad within that domain but bounded by the need for the output contains exactly the values |e_f(0)−e_f(1)| over all binary vertex labelings f with |v_f(0)−v_f(1)|≤1. The standard entry uses finite undirected graphs and the group Z₂; generalizations to other groups or labeling operations define related but distinct invariants.
Clarity¶
The abstraction clarifies a crowded vocabulary by making graph isomorphism preserves the set of attainable edge imbalances because it transports every friendly labeling and its induced edge counts 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 friendly index can mean the scalar produced by one labeling, whereas friendly-index set quantifies over all friendly labelings.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: exponentially many binary labelings, global balance, induced edge parity, graph automorphisms, parity constraints, and attainable-set proofs. Friendly-index set 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: a finite undirected graph with vertices, edges, and binary labels. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the output contains exactly the values |e_f(0)−e_f(1)| over all binary vertex labelings f with |v_f(0)−v_f(1)|≤1 independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of graph theory and graph labeling because they reuse a finite undirected graph with vertices, edges, and binary labels, A balanced vertex labeling induces equal-endpoint and unequal-endpoint edge classes; their count imbalance is computed, and the attainable imbalances are unioned over all friendly labelings., and enumerate or characterize friendly vertex partitions, derive induced edge labels, compute each imbalance, and prove both attainability and exclusion for claimed values. A theorem, diagnostic, or modeling warning can travel when those roles remain literal.
Relationships to Other Abstractions¶
Current abstraction Friendly-index set Domain-specific
Parents (1) — more general patterns this builds on
-
Friendly-index set is a kind of Constraint Prime
The proposed strict upward parent is
prime:constraint.
Hierarchy path (1) — routes to 1 parentless root
- Friendly-index set → Constraint
Neighborhood in Abstraction Space¶
Friendly-index set sits in a crowded region of the domain-specific corpus (20th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Graph Coloring & Labeling (14 abstractions)
Nearest neighbors
- Graceful labeling — 0.94
- Split graph — 0.92
- Bivariegated graph — 0.91
- Zero-symmetric graph — 0.91
- Signed graph — 0.91
Computed from structural-signature embeddings · 2026-09-08