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Investigate the combinatorial properties and configuration counts of the proximity rule P, a local minimax principle minimizing maximum same-state neighbor counts across graph vertices.

August 20, 2026Open Access

The Proximity Rule P: A Minimax Selection Principle on Signed Graphs, with Closed Enumeration on Cycles and a Dichotomy between Complete Graphs and Sparse Meshes (Projective Dynamic Logo Framework — Document D69)

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Investigate the combinatorial properties and configuration counts of the proximity rule P, a local minimax principle minimizing maximum same-state neighbor counts across graph vertices.
Formulated the minimax objective function M(x) = max_v same(v) for binary state assignments on finite simple graphs.
Derived closed enumeration formulas and transfer matrix spectral representations for cycle graph configurations.
Conducted exhaustive computational searches to verify cycle minimizers up to n = 15 and extended evaluations up to n = 31 across graph classes.
Demonstrated that min M = 0 if and only if G is bipartite, while complete graphs yield min M = ceil(n/2) - 1 with C(n, n/2)/2 minimizers forming a single orbit.
Established that cycle minimizers correspond to odd-size matchings with total count T(n) = L(n) + 2 for 3 | n and L(n) - 1 otherwise, governed by the transfer matrix spectrum {φ, -1/φ, ω, conj(ω)}.
Showed that rule P is not switching-invariant and carries c - t independent relation bits on sparse meshes, where c is the cyclomatic number and t is the triangle count.

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Implication

Theoretical analysis reveals exact cycle enumerations and structural dichotomies under the minimax proximity rule, highlighting distinctions between complete graphs and sparse meshes.

Cite This Study

Cédric Laubscher (2026) studied this question.

synapsesocial.com/papers/6a86b6208a91293e6a1cdcfa https://doi.org/10.5281/zenodo.21999711

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Implication

Theoretical analysis reveals exact cycle enumerations and structural dichotomies under the minimax proximity rule, highlighting distinctions between complete graphs and sparse meshes.

The Proximity Rule P: A Minimax Selection Principle on Signed Graphs, with Closed Enumeration on Cycles and a Dichotomy between Complete Graphs and Sparse Meshes (Projective Dynamic Logo Framework — Document D69)

Key Points

Discussion

Implication

Cite This Study

The Proximity Rule P: A Minimax Selection Principle on Signed Graphs, with Closed Enumeration on Cycles and a Dichotomy between Complete Graphs and Sparse Meshes (Projective Dynamic Logo Framework — Document D69)

Key Points

Discussion

Implication

Cite This Study

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