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arXiv · 2603.22546

Axial Morphology of the Partition Graph: Self-Conjugate Axis, Spine, and Concentration

Fedor B. Lyudogovskiy
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Abstract

We study the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary unit transfers between parts. We define the self-conjugate axis, its distance neighborhoods, and the thin spine, a first off-axis layer built from common neighbors of distinct axial vertices. We prove that distinct self-conjugate vertices are never adjacent, that the thin spine is a conjugation-invariant induced subgraph, and that axial and spinal concentration radii differ by at most one. Computations for $1 \le n \le 30$ show that the main local invariants are maximized near the axis and the spine.

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Fedor B. Lyudogovskiy. 2026-03-23. Axial Morphology of the Partition Graph: Self-Conjugate Axis, Spine, and Concentration. https://arxiv.org/abs/2603.22546

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