Spectral radius of $2$-dimensional simplicial complexes with given Betti number

TL;DR

This work determines how the topology of a pure $2$-dimensional simplicial complex, encoded by the Betti number $\beta_2$, constrains its signless Laplacian spectral radius $\mathfrak{q}_1(K)$. By developing an extremal framework for $2$-complexes and leveraging Perron–Frobenius theory, the authors obtain an asymptotic formula $\mathfrak{q}_1(K)=2n-3+\frac{9t}{n^3}+O(n^{-4})$ for large $n$, with extremals realized by the family $\mathcal{T}_n^{2,t}$; in particular, $t=1,2$ yield the unique realizations $\mathsf{T}_n^{2,1}$ and $\mathsf{T}_n^{2,2}$. The results extend spectral extremal methods from graphs to higher-dimensional complexes, anchored by structure theorems for basic holes, path-connectedness, and Euler–Poincaré relations. The techniques combine combinatorial extremal arguments with careful spectral analysis of the signless Laplacian, providing sharp asymptotics and exact low-$t$ extremals that illuminate how topological constraints shape high-dimensional spectra.

Abstract

In this paper we establish an asymptotic formula for the signless Laplacian spectral radius of a $2$-dimensional simplicial complex with given $2$-th Betti number. Furthermore, we characterize the $2$-dimensional simplicial complex that achieves the maximum signless Laplacian spectral radius among all-dimensional simplicial complex with the $2$-th Betti number equal to $1$ or $2$.

Theorems & Definitions (35)

• Lemma 2.1
• proof
• Theorem 3.1: Fan2025hole
• Theorem 3.2: Fan2025hole
• Theorem 3.3: Fan2025hole
• Theorem 3.4: Fan2025hole
• Theorem 3.5: Fan2025hole
• Lemma 3.6
• proof
• Theorem 3.7