Triangle Families with Large Edge Up-Laplacian Spectral Gap

Let $\mathcal{T}$ be a finite nonempty set of $3$-element subsets of a totally ordered set $V$. We view $\mathcal{T}$ as the set of triangles in the support graph. Let $\delta_{1,\mathcal{T}}$ be the signed edge-triangle incidence matrix, and $\lambda(\mathcal{T})$ the spectral gap of $\delta_{1,\mathcal{T}}^T\delta_{1,\mathcal{T}}.$ Our main results show that large $\lambda(\mathcal{T})$ forces strong overlap and a large minimum degree in the support graph. In particular, every support edge lies in at least $\lceil \lambda(\mathcal{T})\rceil-2$ triangles in $\mathcal{T}$ and hence the graph has minimum degree at least $\lceil \lambda(\mathcal{T})\rceil-1$. We further prove that $\binom{n}{3}$ is the exact threshold for attaining level $n:$ if $|\mathcal{T}|< \binom{n}{3}$, then $\lambda(\mathcal{T}) \leq n-1,$ while if $|\mathcal{T}|=\binom{n}{3}$ and $\lambda(\mathcal{T}) > n-1,$ then $\mathcal{T}$ is exactly the full set of triangles on an $n$-vertex clique. Moreover, this clique peak is isolated in a strong interval-scale sense: letting $\phi(t)=\max_{|\mathcal{T}|=t} \lambda(\mathcal{T})$, immediately above $\binom{n}{3}$ there is a forbidden interval on which $\phi(t) \leq n-1$, and the first passage above the level $n-1$ is delayed by $\Theta(n^2)$ additional triangles. Since $\binom{n+1}{3} - \binom{n}{3}=\Theta(n^2),$ this implies that after the peak at $\binom{n}{3}$ one must traverse a nonzero proportion of the full gap until the next clique threshold before substantial recovery can occur. In particular, $\phi$ is not monotone. However, $\phi(t)=\Theta(t^{\frac{1}{3}}).$ Finally, if $\Lambda(t):=\max_{1 \leq s \leq t}\phi(s),$ then $\Lambda(t)=\max\{n \in \mathbb{N}:\binom{n}{3} \leq t\}.$ Thus complete triple systems are the unique minimal spectral extremizers, but their peaks are isolated on the natural scale between consecutive clique thresholds.