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Register a new userThe dual Cheeger constant and spectra of infinite graphs
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In this article we study the top of the spectrum of the normalized Laplace operator on infinite graphs. We introduce the dual Cheeger constant and show that it controls the top of the spectrum from above and below in a similar way as the Cheeger constant controls the bottom of the spectrum. Moreover, we show that the dual Cheeger constant at infinity can be used to characterize that the essential spectrum of the normalized Laplace operator shrinks to one point.
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In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $pto 1,$ we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs.
Given a graph with a designated set of boundary vertices, we define a new notion of a Neumann Laplace operator on a graph using a reflection principle. We show that the first eigenvalue of this Neumann graph Laplacian satisfies a Cheeger inequality.
We investigate self-adjoint extensions of the minimal Kirchhoff Laplacian on an infinite metric graph. More specifically, the main focus is on the relationship between graph ends and the space of self-adjoint extensions of the corresponding minimal Kirchhoff Laplacian $mathbf{H}_0$. First, we introduce the notion of finite and infinite volume for (topological) ends of a metric graph and then establish a lower bound on the deficiency indices of $mathbf{H}_0$ in terms of the number of finite volume graph ends. This estimate is sharp and we also find a necessary and sufficient condition for the equality between the number of finite volume graph ends and the deficiency indices of $mathbf{H}_0$ to hold. Moreover, it turns out that finite volume graph ends play a crucial role in the study of Markovian extensions of $mathbf{H}_0$. In particular, we show that the minimal Kirchhoff Laplacian admits a unique Markovian extension exactly when every topological end of the underlying metric graph has infinite volume. In the case of finitely many finite volume ends (for instance, the latter includes Cayley graphs of a large class of finitely generated infinite groups) we are even able to provide a complete description of all Markovian extensions of $mathbf{H}_0$.
Unfortunately the proof of the main result of [1], Theorem 1, has a flaw. Namely, Lemma 13 used in the proof of Proposition 11 is correct only under an additional assumption that the operator $A$ is normal (adjoint for the one-sided shift operator in $l^2(mathbb N)$ provides a counterexample). Below we prove a version of Lemma 13 that does not require the normality assumption and apply it to prove Proposition 11. In addition, the same version of the lemma appears in paper [2] (as Lemma 3.1) where it is used in the proof of Theorem 1.6. We also explain here how to use the new version of Lemma 13 to correct the proof of Theorem 1.6 from [2].
We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.
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