Cheeger‐like inequalities for the largest eigenvalue of the graph Laplace operator

We define a new Cheeger‐like constant for graphs and we use it for proving Cheeger‐like inequalities that bound the largest eigenvalue of the normalized Laplace operator.


| INTRODUCTION
The (normalized) Laplace operator is a very powerful tool for the study of graphs, as its spectrum encodes important information [8,4,15,2]. Here we consider unweighted and undirected (but oriented) graphs without loops, multiple edges, and isolated vertices. For a fixed such graph on n vertices, let's arrange the n eigenvalues of the Laplace operator, counted with multiplicity, as ≤ ⋯ ≤ λ λ . n 1 We have λ = 0 1 , and the multiplicity of the eigenvalue 0 equals the number of the connected components of the graph. Thus, (1) if and only if the graph is connected. Henceforth, we shall only consider connected graphs. There is also a quantitative aspect. As we shall explain in more detail below, λ 2 estimates the coherence of the graph, that is, how different it is from a disconnected one.
The largest eigenvalue, which is the main object of interest of this article, satisfies with equality if and only if the graph is complete. For noncomplete gaphs ≥ λ n n + 1 − 1 , n with equality if and only if the graph either is obtained from a complete graph by removing a single edge or consists of two complete graphs of size n + 1 2 that share a single vertex [16]. In the other direction with equality if and only if at least one connected component of the graph is bipartite. For connected graphs, the first nonzero eigenvalue λ 2 is controlled both above and below by the Cheeger constant h, a quantity that measures how difficult it is to partition the vertex set into two disjoint sets V 1 and V 2 such that the number of edges between V 1 and V 2 is as small as possible and such that the volume of both V 1 and V 2 , that is, the sum of the degrees of their vertices, is as big as possible. In particular, Furthermore, there is an interesting characterization of h obtained by writing λ 2 using the Rayleigh quotient and then replacing the L 2 -norm by the L 1 -norm both in the numerator and denominator, as we shall see in Section 2.
In this paper, we want to explore an analogue of this for λ n . In the same sense that by (3), λ 2 estimates how different a connected graph is from being disconnected, by (2), λ 2 − n should quantify how different the graph is from being bipartite. One might therefore try to find the best (in a suitable sense) bipartite subgraph of our graph, because for a bipartite graph, the Rayleigh quotient that we shall discuss below is 2, the maximal possible value. In fact, as it turns out, that subgraph can be quite small. More precisely, we shall introduce a new constant that is an analogue of the Cheeger constant in the sense that it can be characterized by writing λ n using the Rayleigh quotient and then replacing the L 2 -norm by the L 1 -norm both in the numerator and denominator. This constant is very simple, nalogously to the Cheeger estimate (3), we shall prove that it controls the largest eigenvalue λ n both above and below. Therefore, Q is an analogue of the Cheeger constant for the largest eigenvalue. As we had explained above, λ 2 controls how different the graph is from a connected. Analogously, in view of (2), one should expect that λ 2 − n measures the difference from a bipartite graph.
Throughout the paper we shall also prove new general results of spectral graph theory that are useful to prove or discuss our main result.

| Structure of the paper
In Section 2 we discuss the Laplace operator, the Cheeger constant, the dual Cheeger constant and the edge-Laplacian, as preliminaries to our work. In Section 3, and in particular in JOST AND MULAS | 409 Theorem 3, we present our main results and we prove them in Section 4. In Section 5 we motivate the choice of Q, in Section 6 we discuss the precision of our lower bound for λ n , and finally in Section 7 we discuss the precision of our upper bound.

| PRELIMINARIES
In this section, we present some well-known results of spectral graph theory as preliminaries to our work; a general reference is [8].
From here on we fix a graph V E Γ = ( , ) on n vertices. We also fix an arbitrary orientation on Γ, that is, we see each edge as an arbitrarily ordered pair of its endpoints. Given ∈ e v w E = ( , ) , we say that v is the input of e and w is its output. The fixed orientation is needed to do the computations when considering a function  → γ E : . However, the results are independent of the chosen orientation because, if one reverses the orientation of some edges, changing the sign of γ on these edges leads to the same results. Therefore, the oriented edges considered here should not be confused with directed edges. Moreover, we shall use the notation v w for indicating (oriented) edges when input and output do not need to be distinguished.

| Laplace operator and its eigenvalues
Let Id be the n n × identity matrix, let A be the adjacency matrix of Γ, let D be the diagonal degree matrix and let by the min-max principle, the eigenfunctions of the other eigenvalues must be orthogonal to them with respect to the scalar product The orthogonality to the constants is satisfied also by the eigenfunctions of λ n , but in this case we do not need to specify it.

| Cheeger constant
For a connected graph V E Γ = ( , ), the Cheeger constant is defined as V S E S S ,¯, ( ,¯) denotes the number of edges with one endpoint in S and the other in S, and . The following theorem [1,12] gives two important bounds for λ 2 in terms of h. Theorem 1. For every connected graph, Also, the following theorem [8, Theorem 2.8 and Corollary 2.9] shows the interesting relation between h and λ 2 when, in the characterizations of λ 2 via the Rayleigh quotient, we replace the L 2 -norm by the L 1 -norm both in the numerator and denominator.
Our Theorem 3 is an analogue of Theorems 1 and 2 for the largest eigenvalue λ n in terms of our new constant Q. Before stating it, we shall discuss the dual Cheeger constant and the edge-Laplacian.

| Dual Cheeger constant
In literature, there is already a Cheeger-like constant that bounds the largest eigenvalue [3,5]. It is defined as it is called the dual Cheeger constant and it satisfies an analogue of (4), The two constants h and h are actually related to each other [5]. For the dual Cheeger constant, however, there is no result analogous to Theorem 2 [10]. This motivates the definition of the new constant Q that again bounds λ n and, additionally, satisfies an analogue of Theorem 2.

| Edge-laplacian
Associated to the Laplace operator there is also the edge-Laplacian, defined as Instead on acting on functions defined on the vertex sets, L E acts on functions defined on the edge set. It has the same nonzero spectrum of L (i.e., the nonzero eigenvalues are the same, counted with multiplicity) and the multiplicity of the eigenvalue 0 for L E equals the number of cycles of Γ [15]. We can therefore write the largest eigenvalue (that coincides for L and L E ) also in terms of the Rayleigh quotient for functions on the edge set, by applying the min-max principle to L E : , 0 In Section 3 we shall present an analogue of Theorems 1 and 2, where: • We look at λ n instead of λ 2 .
• We use Q instead of h.
• We use the point of view of the edge-Laplacian for considering the Rayleigh quotient and characterize Q.

| MAIN RESULTS
Before stating our main theorem, let's recall that for a graph Γ we have defined the new Cheeger-like constant Let's also define the constant Observe that the characterization of Q appearing in Theorem 3 equals the Rayleigh quotient we have used for writing λ n from the point of view of the edge-Laplacian, replacing the L 2 -norm by the L 1 -norm. Therefore, such a characterization is analogous to the one of h in Theorem 2. We prove Theorem 3 in Section 4. Also, in Section 5 we motivate the choice of Q, in Section 6 we discuss whether the lower bound appearing in Theorem 3 is sharp, and in Section 7 we discuss the sharpness of the upper bound.

| PROOF OF THE MAIN RESULTS
We split the statement of Theorem 3 into three parts. The first part, Lemma 4, contains the characterization of Q. The second part, Lemma  Proof. To prove that , 0 and let  → γ E ′: be 1 on v v ( , ) 1 2 and 0 otherwise. Then,  . Then, This proves the claim. □ As a corollary of Lemma 4, we get another characterization of Q.  . Then, by taking ⊂ Γ Γ as the bipartite graph containing only the edge v v ( , ) 1 2 , we get that  If there is no such vertex, then

| Lower bound for the largest eigenvalue
Therefore, the bound in Lemma 6 is better than the usual bound ≤ λ n n n − 1 only for a small class of graphs. However, the aim of our work is not to find the best possible bounds forλ n but the best possible bounds for λ n involving Q, to show that Q is a Cheeger-like constant. We shall see, in Section 6, that the bound in Lemma 6 is actually the best possible lower bound for λ n involving only Q.

| Upper bound for the largest eigenvalue
Lemma 7. For every graph, Proof. We apply [18,Theorem 5] to obtain Observe that the bound in Lemma 7 is not a better upper bound for λ n than the one in [18,Theorem 5]. Nevertheless, it is a good upper bound for λ n involving Q, as we shall see in Section 7.

| CHOICE OF Q
Let us motivate the choice of Q. As we have discussed in Section 2, We have chosen Q to be the constant that can be written as (6) by replacing the L 2 -norm by the L 1 -norm both in the numerator and denominator. We could have chosen to work on the constant that can be written as (5) by replacing the L 2 -norm by the L 1 -norm, but such a constant is actually equal to 1 for all graphs, as shown by the following lemma. Furthermore, while the characterization of the Cheeger constant is interesting also because it is equal to the second smallest eigenvalue of the 1-Laplacian, one cannot get an analogous constant in this sense because the largest eigenvalue of the 1-Laplacian equals 1 for every graph, as shown in [8,Theorem 5.1]. For completeness, we shall provide a proof.
Lemma 8. For every graph, and assume, without loss of generality, that To see the inverse inequality, let  → f Ṽ : that is 1 on a fixed vertex and 0 on all other vertices. Then, , 0 □ 6 | HOW GOOD IS THE LOWER BOUND?
To see that ≤ Q λ n is a sharp lower bound, consider the case of For further motivating our upper bound, we shall: (1) Prove that, for each graph on n nodes, ⋅ τ n < 0.54 and 0.54 is the best ε with a precision of two decimal places such that if c is a constant that does not depend on n, as we might be tempted to do by looking at the example of regular graphs.
To prove these two points, we shall first discuss one-sided bipartite graphs, a new big class of graphs that includes among others petal graphs, complete graphs, and complete bipartite graphs.

| One-sided bipartite graphs
Definition Fix n and k such that Call such a graph a k d ( , )-one-sided bipartite graph.
Remark 5. In a k d ( , )-one-sided bipartite graph, the vertex set is divided into two sets V 1 and V 2 . All possible edges between V 1 and V 2 are there, the k vertices in V 2 are not connected to each other and the vertices in V 1 all have degree d, therefore there are edges between vertices of V 1 if and only if d k > (Figure 1). In particular, a k d ( , )-one-sided bipartite graph is: • The petal graph if k = 1 and d = 2.
• The complete graph K n if k = 1 and d n = − 1.
• The graph Proof. This follows easily by definition of one-sided bipartite graphs and by [11,Theorem 2.6], which states that a d-regular graph on n nodes exists if and only if at least one of d and n is even. □ In Theorem 11 we shall prove that for a one-sided bipartite graph with ≥ d n k − , λ d k d = + n and for a k d ( , )-one-sided bipartite graph with d n k < − , n Let's prove a preliminary lemma first.
Definition [4]. Given a vertex v 1 , let  ⊂ v V ( ) 1 be the set of neighbors of v 1 . We say that v 1 and v 2 are duplicate vertices if   v v ( ) = ( ) 1 2 .
Observe that, in particular, duplicate vertices have the same degree and they cannot be neighbors of each other.