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.

Version: v2 · 2026-09-19 · History
Domain-specific #: 1889
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Graph Labeling → Mathematics

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.

How would you explain it like I'm…

Fair Coloring Difference List

Draw some dots and connect them with lines. Color each dot black or white, using almost the same number of each. Each line is a 'same' line if both ends match, or a 'different' line if they don't. Count how far apart the numbers of same and different lines are, then try every fair coloring and collect all the answers you can get. That collection is the friendly-index set.

All the Balanced-Label Gaps

A graph is a set of dots (vertices) joined by lines (edges). Label each dot 0 or 1 so the number of 0s and 1s differ by at most one; that is a 'friendly' labeling. Each line gets a label too: 0 if its two ends match and 1 if they differ. Then count how many lines are 0 and how many are 1, and take the difference without a minus sign. Do this for every friendly labeling, and collect all the differences you find. That collection is the friendly-index set.

Friendly Labeling Imbalance Set

In graph labeling, a binary vertex labeling is friendly if the number of vertices labeled 0 and labeled 1 differ by at most one. Each edge then gets an induced label: the sum of its endpoint labels mod 2, which is 0 when the endpoints match and 1 when they differ. For one labeling you compute |e(0) − e(1)|, the absolute difference between the numbers of 0-edges and 1-edges. The friendly-index set is the set of all these values over every friendly labeling, not just the value for one. To claim a value belongs to the set you must show a labeling that achieves it, and to rule a value out you must prove no friendly labeling can.

For a graph G, a binary vertex labeling f is friendly if |v_f(0) − v_f(1)| ≤ 1, where v_f(i) counts vertices labeled i. Each edge uv receives the induced label f(u) + f(v) mod 2, splitting edges into equal-endpoint (label 0) and unequal-endpoint (label 1) classes with counts e_f(0) and e_f(1). The friendly-index set of G is the set of all values |e_f(0) − e_f(1)| as f ranges over every friendly labeling. It is a graph invariant built from four pieces: a vertex-balance constraint, induced (not independently chosen) edge labels, an absolute imbalance statistic, and quantification over all admissible labelings. Establishing it requires proving both attainability of each claimed value and exclusion of others. It connects to cordial labelings, where small imbalance is attained, and to full-cordiality behavior, and is studied by deriving the attainable imbalances for families of graphs.

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.
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) — outgoing typed relations

Friendly-index set is a kind of Constraint Prime

Friendly labelings literally satisfy a global balance constraint and the invariant is generated over that constrained family.

Hierarchy path (1) — routes to 1 parentless root

Friendly-index set → is a kind of Constraint

Neighborhood in Abstraction Space¶

Friendly-index set sits in a crowded region of the domain-specific corpus (11^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Graph Classes & Invariants (40 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 · 2026-10-25