2024/09/11 by Chuanyuan Ge, Ge, Chuanyuan
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2409.07097
openalex publication_date 2024/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
As a non-trivial extension of the celebrated Cheeger inequality, the higher-order Cheeger inequalities for graphs due to Lee, Oveis Gharan and Trevisan provide for each k an upper bound for the k-way Cheeger constant in forms of C(k)√(λk(G)), where λk(G) is the k-th eigenvalue of the graph Laplacian and C(k) is a constant depending only on k. In this article, we prove some new bounds for multi-way Cheeger constants. By shifting the index of the eigenvalue via cyclomatic number, we establish upper bound estimates with an absolute constant instead of C(k). This, in particular, gives a more direct proof of Miclo's higher order Cheeger inequalities on trees. We also show a new lower bound of the multi-way Cheeger constants in terms of the spectral radius of the graph. The proofs involve the concept of discrete nodal domains and a probability argument showing generic properties of eigenfunctions.