Faber-Krahn inequality and heat kernel estimates on glued graphs

TL;DR

This work develops a discrete framework to study Faber-Krahn inequalities and heat kernels on graphs built by gluing several pages to a spine, potentially along infinite sets of vertices. By defining relative Faber-Krahn functions and proving a gluing lemma, the authors bound the glued graph’s FK function in terms of its pages and translate these bounds into heat-kernel estimates, with explicit forms depending on page volumes. In book-like graphs with a transient spine, they obtain matching two-sided heat-kernel bounds on spine-to-spine transitions, supported by uniformity and Harnack-type hypotheses; lower bounds emerge from parabolic Harnack inequalities and transience via $h$-transforms. The results provide a discrete counterpart to continuous ends-manifolds surgery, offering a blueprint for sharp HK bounds in glued settings and laying groundwork for more general two-sided estimates in future work. These findings have potential implications for understanding diffusion on complex networks formed by gluing simpler geometric pieces.

Abstract

Faber-Krahn functions provide lower bounds on the first Dirichlet eigenvalue of the Laplacian and are useful because they imply heat kernel upper bounds. In this paper, we are interested in Faber-Krahn functions and heat kernel estimates for a certain class of graphs consisting of "sufficiently nice pages" (satisfying a Harnack inequality) glued together via a "sufficiently nice spine." For such graphs, we obtain a relative Faber-Krahn function in terms of the Faber-Krahn functions on the pages. The corresponding heat kernel upper bound involves the volumes on the various pages. In the case our graphs satisfy a property we call "book-like" and the spine is appropriately transient, we provide a matching lower bound for the heat kernel between two points on the gluing spine.

Figures (2)

• Figure 1: This figure illustrates gluing two half-planes with lazy SRW along a copy of $\mathbb{Z}$ as described in Example \ref{['glue_half']}. The numbers shown are vertex weights. On the left, the orange spine vertices have weight zero, so it is not shown. All edges have weight one. On the right, the "doubled" edges from the gluing now have weight two, while all other edges retain weight one.
• Figure 2: Begin with the lazy simple random walk on $\mathbb{Z}^2$ in the top left. Vertex weights are shown, and all edges have weight one. We cut it apart by removing the set of vertices where $y=0$, leaving us the two graphs with trailing edges on the top right. (The "removed" $\Gamma_0$ vertices are treated as a disconnected set all with weight zero, and are not shown.) By capping the trailing edges with vertices, adding back edges between these vertices, and taking the Neumann random walk, we obtain the bottom left pair of graphs with given vertex weights and all edge weights one. Compare with Figure \ref{['gluing_ex']}. The pair of graphs on the bottom right, where all unlabeled edges have weight one, can be glued back together into $\mathbb{Z}^2$ with the lazy simple random walk.

Theorems & Definitions (71)

• Definition 1: Controlled weights
• Definition 2: The notation $\approx$
• Definition 3: Uniformly lazy
• Definition 4: (Lazy) Simple random walk
• Definition 5: Boundary of a subgraph
• Remark 1
• Lemma 1
• proof
• Lemma 2
• Remark 2