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

TL;DR

This paper explores the structure of the partition graph, introducing concepts like the self-conjugate axis and thin spine, and analyzes their properties and invariants through theoretical proofs and computational experiments.

Contribution

It defines new structural features of the partition graph and proves key properties, enhancing understanding of its geometric and combinatorial characteristics.

Findings

Self-conjugate vertices are never adjacent.

The thin spine is conjugation-invariant.

Concentration radii differ by at most one.

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.