Projects
1) Triangle selection to maximize $\lambda_{\min}^+$
Fix a graph and choose a subset of its triangles in order to optimize $\lambda_{\min}^+(L_1(T))$. This raises:
- Monotonicity: which pieces are monotone under adding triangles?
- Perturbation: how much can one triangle change the bottom of the spectrum?
- Saturation: when do additional triangles stop helping the total operator?
Key structural input
Triangle addition produces a rank-one PSD update on $L_1^{\uparrow}$: $$L_1^{\uparrow}(T\cup\{\tau\})=L_1^{\uparrow}(T)+b_\tau b_\tau^{\mathsf T}.$$ This invites greedy heuristics and interlacing-based certificates.
2) Structured spectra (SRGs, designs)
For highly symmetric graphs, one can sometimes reduce $L_1^{\uparrow}$ to small blocks and compute spectra exactly. This is a natural place for rigorous theorems and clean examples.
3) Computation-as-theorem-finder
Enumerate small graphs, detect monotonicity failures or extremizers, then reverse-engineer a clean invariant or hypothesis.