The Degree Landscape of the Partition Graph: Maximal Degree, Extremal Vertices, and Spectra

TL;DR

This paper analyzes the degree landscape of the partition graph, providing exact formulas for maximal degree, extremal vertices, and spectral properties, supported by computational data for small n.

Contribution

Introduces the degree layers, spectrum, and invariants of the partition graph, with exact formulas for maximal degree and extremal classifications, advancing understanding of the graph's structure.

Findings

Exact formula for the maximal degree of the partition graph.

Maximal-degree vertices lie on the maximal-support stratum.

Computational data for 1 ≤ n ≤ 60 illustrating extremal properties.

Abstract

We study the degree landscape of the partition graph $G_n$, whose vertices are the integer partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts, followed by reordering. Using the previously established local degree formula, we introduce the degree layers $D_d(n)$, the degree spectrum $Spec_D(n)$, and the numerical invariants $\Delta_n$, $m_\Delta(n)$, and $s(n)$. The main theorem provides an exact formula for the maximal degree. If $$ \rho(n):=\max\{r:T_r\le n\},\qquad T_r=\frac{r(r+1)}{2}, $$ and $$ \nu:=n-T_{\rho(n)}, $$ then $$ \Delta_n=\rho(n)\bigl(\rho(n)-1\bigr)+\beta_{\rho(n)}(\nu), $$ where $\beta_r$ is an explicit budget function governed by a square--pronic threshold rule. We also prove that every maximal-degree vertex lies on the maximal-support stratum, and we obtain exact extremal classifications at the levels $n=T_t$, $n=T_t+1$, and $n=T_t+2$. The paper also includes a finite computation on the range $1\le n\le 60$, recording extremal multiplicities, representative extremal shapes, spectrum sizes, selected degree histograms, and first data on contact between the extremal layer and the self-conjugate axis. This computational part is deliberately limited in scope. It is descriptive rather than exhaustive, and is included only as a first numerical profile of the degree landscape.