Dirichlet -Laplacian eigenvalues and Cheeger constants on symmetric graphs
In this paper, we study eigenvalues and eigenfunctions of -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 -Laplacian, as 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.
The spectrum of the Laplacian on a domain in the Euclidean space was extensively studied in the literature, see e.g. [CH53, RS78]. There were many far-reaching generalizations on Riemannian manifolds, see [Cha84, SY94].
In 1970, Cheeger [Che70] introduced an isoperimetric constant, now called Cheeger constant, on a compact manifold to estimate the first non-trivial eigenvalue of the Laplace-Beltrami operator, see also [Yau75, KF03]. A graph is a combinatorial structure consisting of vertices and edges. Cheeger’s estimate was generalized to graphs by Alon-Milman [AM85] and Dodziuk [Dod84], respectively. Inspired by these results, there were many Cheeger type estimates on graphs, see e.g. [DK86, Lub94, Fuj96, LOGT12, BHJ14, Liu15, BKW15, KM16, TH18]. It turns out that Cheeger’s estimates on graphs are useful in computer sciences [DH73, NJW, Bol13].
As elliptic operators, the -Laplacians are nonlinear generalizations of the Laplacian in Euclidean spaces and Riemannian manifolds. The spectral theory of the -Laplacians were studied by many authors, to cite a few [Lin93, Mat00, Wan12, Val12, AC13, NV14, SW17]. Yamasaki [Yam79] proposed a discrete version of -Laplacian on graphs. The spectral theory for discrete -Laplacians was studied by [Amg03, Tak03, HB10, CSZ15, KM16]. It was well-known that the Cheeger constant is equal to the first eigenvalue of -Laplaican, i.e. a Sobolev type constant, see [FF60, CO97, Chu97, Li12, CSZ15, Cha16, KM16, CSZ17b]. So that Cheeger’s estimate reveals a connection between the first eigenvalues of -Laplacians for . In this paper, we study the spectral theory of -Laplacians on graphs, and use the limit to investigate the Cheeger constant.
We recall the setting of weighted graphs. Let be a locally finite, simple, undirected graph. Two vertices are called neighbours, denoted by , if there is an edge connecting and i.e. Let be the edge weight function. We extend to by setting if Let be the vertex measure. The weights and can be regarded as discrete measures on and respectively. We call the quadruple a weighted graph. For any subset we denote by the norm of a function on
For a weighted graph and a finite subset we define the -Laplacian, , with Dirichlet boundary condition on We denote by the set of functions on For any function the null-extension of is denoted by i.e. and otherwise. The -Laplacian with Dirichlet boundary condition, Dirichlet -Laplaican in short, on is defined as
We say is an eigenfunction (or eigenvector) pertaining to the eigenvalue for the Dirichlet -Laplacian on if
For any let be the -Dirichlet functional on defined as
As is well-known, for is an eigenfunction for the Dirichlet -Laplacian on if and only if is a critical point of the functional under the constraint The critical point theory for the case is subtle, see e.g. [HB10, Cha16], for which the operator is called -Laplacian. Note that the -Laplacian depends on the weights and If we choose then the associated -Laplacian is called normalized -Laplacian. The -Laplacian is a linear operator if and only if
In this paper, we are interested in the first eigenvalue (the maximum eigenvalue resp.), i.e. the smallest (largest resp.) eigenvalue, denoted by ( resp.), and the associated eigenfunctions for -Laplacians, . By the well-known Rayleigh quotient characterization,
Analogous to the continuous case [KL06], we obtain the following characterization of first eigenfunctions. A finite subset is called connected if the induced subgraph on is connected, i.e. for any two vertices in there is a path in the induced subgraph on connecting them.
Let be a weighted graph and be a finite connected subset of Then the eigenfunction of Dirichlet -Laplacian on is a first eigenfunction if and only if either on or on . Moreover, the first eigenfunction is unique up to the constant multiplication.
The first eigenfunction can be characterized via the fixed-sign condition. The uniqueness of the first eigenfunction will be crucial for our applications.
For any we denote by the edge boundary of The (Dirichlet) Cheeger constant on a finite subset is defined as
where and . See Definition 5.3 for Cheeger constants, and of infinite graphs. The subset attains the minimum in (5) is called a Cheeger cut of For a finite graph without boundary the Cheeger constant was proven to be equal to the first nontrivial eigenvalue of -Laplacian, see e.g. [HB10, Proposition 4.1] and [Cha16, Theorem 5.15]. The following is an analogous result for the Dirichlet boundary case.
Let be a finite subset of Then
where is the first eigenvalue of Dirichlet -Laplacian.
For the linear normalized Laplacian on a finite graph without boundary, the maximum eigenvalue can be used to characterize the bipartiteness of the graph. Recall that a graph is called bipartite if its vertex set can be split into two subsets such that every edge connects a vertex in to one in As is well-known [Chu97], the maximum eigenvalue is if and only if the graph is bipartite. More importantly, there is an involution
which transfers an eigenfunction of eigenvalue to an eigenfunction of eigenvalue Similar results hold for linear normalized Laplacians with Dirichlet boudnary condition, see [BHJ14]. By this result, one easily figure out the sign condition for the maximum eigenfunction via that of the first eigenfunction. However, the involution doesn’t work well for the nonlinear case, i.e. see e.g. Example 4.1. By using a convexity argument, we circumvent the difficulty and give the characterization of maximum eigenfunction by the sign condition for bipartite subgraphs.
Let be a weighted graph and be a finite connected bipartite subgraph of . Assume is an eigenfunction of Dirichlet -Laplacian on . Then is a maximum eigenfunction if and only if satisfies for and . Moreover, the maximum eigenfunction is unique up to the constant multiplication.
We study Cheeger constants on symmetric graphs. For a weighted graph an automorphism of is a graph isomorphism satisfying
The set of automorphisms of form a group, denoted by For our purposes, we say that an infinite graph is “symmetric” if there is a subgroup of the automorphism group acting on finitely, i.e. each orbit for the action of the group called -orbit, consists of finitely many vertices. For any we denote by the -orbit of We define the quotient graph as follows: The set of vertices consists of the -orbits; two different orbits are adjacent if there are such that and the edge weight is defined as
the vertex weight is defined as Note that in our definition, the quotient graph has no self-loops, although there could be edges between vertices in one orbit in .
Let be a weighted graph, and be a subgroup of the automorphism group which acts finitely on Then
This theorem yields that we can reduce the computation of Cheeger constants of to that of the quotient graph.
Note that the Cheeger cuts of a graph are usually not unique, see e.g. Example 5.1. But the proof of theorem indicates that among them there is one Cheeger cut consisting of -orbits. So that, for the computation of Cheeger constants of a symmetric graph one can treat the orbits as integrality.
There is a natural metric on the graph , the combinatorial distance , defined as , i.e. the length of the shortest path connecting and by assigning each edge the length one. For the combinatorial distance of the graph, we denote by the ball of radius centered at and by the -sphere centered at . We call a graph is spherically symmetric centered at a vertex if for any and , there exits an automorphism of which leaves invariant and maps to , see [KLW13, BK13]. Then there is an associated subgroup acts finitely on such that the -orbits are exactly . In this case, the quotient graph is a “one dimensional” model. In , we denote by the ball of radius centered at and by the annulus of inner radius and outer radius
Let be a spherically symmetric graph centered at with the associated subgroup of the automorphism group. Then
Moreover, if has infinite -measure, then
The paper is organized as follows. The basic set up and concepts introduced in §2. In §3, we prove the sign characterization of first eigenfunctions, Theorem 1.1. In §4, we prove Theorem 1.2, the sign characterization of maximum eigenfuntions for bipartite subgraphs. In §5, using the analytic approach, we identify the Cheeger constant of a symmetric graph with that of the quotient graph, Theorem 1.3. In §6, we introduce a “one dimensional” model graph as the quotient graph of a spherically symmetric graphs, and prove Theorem 1.4. In Appendix, we calculate various Cheeger constants of spherically symmetric graphs, for example, Fujiwara’s spherically symmetric trees in Appendix A.1 and Wojciechowski’s anti-trees in Appendix A.2.
For a weighted graph and a finite subset we define the -Laplacian, , with Dirichlet boundary condition on We denote by the set of functions on For any function the null-extension of defined as
Throughout the paper, we denote the null-extension of any function by in the paper. The -Laplacian with Dirichlet boundary condition, Dirichlet -Laplaican in short, on is defined as
For any we denote the space w.r.t. the measure by
and denote the -norm of a function by
The difference operator is defined by for any . Then for any , is a function on given by , . We denote the -norm of the function by
For convenience, we omit the subscript if it is clear in the context, e.g. and so on.
3. First eigenfunctions and eigenvalues to Dirichlet p-Laplacians
In this section, we give an equivalent characterization for first eigenfucntions of Dirichlet -Laplacian. Firstly, we prove the following lemma.
Let be a weighted graph and be a finite connected subset. Assume is an eigenfunction of Dirichlet -Laplacian on . If (, resp.) on , then ( resp.) on .
We show this by contradiction. Suppose and there exists such that . By (2), we have
Hence, for any By the connectedness of , on This contradicts to since is an eigenfunction. Hence, .
Replacing by and using the same argument, we can show that gives . ∎
For any , we denote the vertex boundary of by
Let be functions on . Assume satisfy the following equation
Then on .
The comparison principle enables us to characterize the first eigenfunctions by their sign conditions.
Let be a finite connected subgraph of weighted graph and be an eigenfunction of Dirichlet -Laplacian on . Then is the first eigenfunction if and only if either on or on .
We first show that the first eigenfunction satisfies either on or on . Let be a first eigenfunction pertaining to . By scaling, w.l.o.g., we assume that By the Rayleigh quotient characterization (9), we have
Hence, above inequalities are equalities. This implies that
For another direction, we choose as the positive first eigenfunction pertaining to the first eigenvalue , replacing by if . Using a contradiction argument, we assume that is a positive eigenfunction of Dirichlet -Laplacian pertaining to with . Since is finite, by scaling we may assume that for any We claim that with Note that on
and The comparison principle for the -Laplacian, Lemma 3.2, yields that on This proves the claim.
By the same argument, replacing by one can show that on for any Taking the limit we get on This yields a contradiction. Hence, we obtain and is the first eigenfunction. ∎
With the same assumption as in Theorem 3.1, the first eigenfunction is unique (up to multiplication with constants).
Let be two first eigenfunctions of Dirichlet -Laplacian. It suffices to prove there exists a constant such that .
By Theorem 3.1, either on or on for . We may assume on and for . Choose a new function Then Let , , be the null-extension of , , , respectively. We claim that
If or then the equality holds trivially. It’s sufficient to prove the claim for the case This follows from the convexity of norm, denoted by , in . Indeed, setting vectors and we have
Hence the above inequalities are in fact be equalities, which implies that for any with This yields that on by the connectedness of The proof is completed. ∎
Proof of Proposition 1.1.
Let be characteristic function on defined by
For any function , let and . Set . Then if and only if . Hence,
Since , . By (13),
We also have if and only if . Then
On the other hand, let be a Cheeger cut such that . Considering the characteristic function , by (4), we have
Hence, we obtain . The proof is completed. ∎
In the rest of the section, we prove the monotonicity property of the first eigenvalue of Dirichlet -Laplacian as varies, analogous to the continuous case. By mimicking the argument in [Lin93, Theorem 3.2], we prove the following result.
Let be a finite connected subset of a weighted graph with . For we have
Let be a first eigenfunction of Dirichlet -Laplacian on satisfying on and Then Let Then By the Rayleigh quotient characterization (9),
Let For any given , by the symmetry , we always assume that ,
where denotes Noting that and applying Hölder inequality, we obtain
From the Hölder inequality, we have
Combining the above two inequalities, together with , we have
4. Maximum eigenfunctions to Dirichlet p-Laplacians on bipartite subgraphs
Recall that a graph is called a bipartite graph if its vertices can be divided into two disjoint sets and such that every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. In this section, we obtain an equivalent characterization for maximum eigenfunctions on a bipartite subgraph.
Let be the involution defined as (6). Now we give an example to show the relationship between first eigenfunctions and maximum eigenfunctions for Dirichlet -Laplacian when .
Let be a weighted graph with as shown in Figure 1. Assume for and , for . For , by direct computation, the first eigenfunction and maximum eigenfunction for , , are
Obviously, . This example indicates that there is no close relation between first and maximum eigenfunctions for .
Next we describe the sign property of maximum eigenfunctions for -Laplacian on a bipartite subgraph .
Let be a weighted graph and be a finite connected bipartite subgraph of . If is a maximum eigenfunction of Dirichlet -Laplacian on , then satisfies for and .
Firstly we assume that is a maximum eigenfunction satisfying . It’s sufficient to show that satisfies for and . Let with the involution defined as (6), and , be null-extension of , defined as (7), respectively. Since is a connected bipartite subgraph, then
Hence, we have
Then the inequalities in (18) have to be equalities, which implies that
By (20), it suffices to show that there is no vertex in such that . We show this by contradiction. Suppose there is such that , then . By Green’s formula (ref. [Gri09]), (19) yields that is an eigenfunction satisfying eigen-equation (2). By (8),
By is a connected bipartite subgraph with bipartite parts and ,
To prove the other direction of Theorem 1.2, we need the some lemmas.
We write and the vertex boundary
For any function , let . For simplicity, we write for . Then the conditions and are given by and , respectively. Hence, restricted corresponds to a vector .