## 1 Introduction

In their recent beautiful work on the equiangular lines problem, Jiang, Tidor, Yao, Zhang, and Zhao [JTY+19] proved the following remarkable result.

###### Theorem 1.1.

If is a connected graph of maximum degree on vertices, then the multiplicity of the second largest eigenvalue of its adjacency matrix is bounded by

For their application to equiangular lines, [JTY+19] only needed to show that the multiplicity of the second eigenvalue is , but they asked whether the dependence in Theorem 1.1 could be improved, noting a huge gap between this and the best known lower bound of achieved by certain Cayley graphs of (see [JTY+19, Section 4]). Apart from Theorem 1.1, there are as far as we are aware no known sublinear upper bounds on the second eigenvalue multiplicity for any general class of graphs, even if the question is restricted to Cayley graphs (unless one imposes a restriction on the spectral gap; see Section 1.1 for a discussion). As this eigenvalue plays an important role in many areas of mathematics, any general phenomena concerning it are of potentially broad impact.

In this work, we prove significantly stronger upper bounds under the additional assumption that the graph is regular. Graphs are undirected and allowed to have multiedges and self-loops unless explicitly stated as being simple. Order the eigenvalues of as , and let denote the number of eigenvalues of in an interval .

###### Theorem 1.2.

If is a connected regular graph on vertices with , then^{4}^{4}4All asymptotics are as and the notation suppresses terms.

(1) |

If is simple, then further

(2) |

In addition to the improved bounds, one difference between our results and Theorem 1.1 is that we control the number of eigenvalues in a small interval containing . Though we do not know whether the exponents in (1), (2) are sharp, we show in Section 5.1 that constant degree bipartite Ramanujan graphs have at least eigenvalues in the interval appearing in (1), indicating that is the correct regime for the maximum number of eigenvalues in such an interval when is a constant. Note that as for a connected graph by [NIL91], the width of the interval under consideration increases with the degree.

A second difference is that we obtain nontrivial bounds for all assuming the graph is simple, whereas Theorem 1.1 requires . As remarked in [JTY+19], Paley graphs have degree and second eigenvalue multiplicity , so some bound on the degree is required to control the multiplicity. It is also clear that the multiplicity can be if the graph is disconnected, by taking many copies of a small graph.

A closed walk of length in a graph is a sequence such that and each is an edge in . We refer to the vertex as the root of the walk, and to the number of distinct vertices in as its support. The main ingredient in the proof of Theorem 1.2 is a new lower bound on the typical support of a random closed walk in a regular graph, rooted at any vertex.

###### Theorem 1.3.

Suppose is connected and regular, is any vertex in , and is a uniformly random closed walk in of length rooted at . Then

(3) |

If is simple, then moreover

(4) |

It may be tempting to compare Theorem 1.3 with the familiar fact that a random closed walk of length on (or in continuous time, a standard Brownian bridge run for time attains a maximum distance of from its origin. However, as seen in Figure 1, there are regular graphs for which a closed walk of length from a particular vertex travels a maximum distance of only

with high probability. Theorem

1.3 reveals that nonetheless the number of distinct vertices traversed is always typically . We show that regularity is in fact necessary for this phenomenon by analyzing in Section 5.2 an irregular “lollipop” graph for which the typical support of a closed walk from a specific vertex^{5}

^{5}5It remains plausible that an averaged version of Theorem 1.3 in which is random holds for irregular graphs. is only . We do not know if the specific exponents of and appearing in Theorem 1.3 are sharp, but considering a cycle graph shows that it is not possible to do better than .

Given Theorem 1.3, our proof of Theorem 1.2 follows the strategy of [JTY+19]: since most closed walks in have large support, the number of such walks may be drastically reduced by deleting a small number of vertices from

. By a moment calculation relating the spectrum to closed walks and a Cauchy interlacing argument, this implies an upper bound on the multiplicity of

. The crucial difference is that we are able to delete only vertices whereas they delete .The key ingredient in our proof of Theorem 1.3 is a result regarding the Perron eigenvector (i.e., the unique, strictly positive eigenvector with eigenvalue ) of an irregular connected graph, which may be of independent interest.

###### Theorem 1.4.

Let be an irregular connected graph of maximum degree with at least two vertices, and let be the

-normalized Perron vector of

. Then there is a vertex with degree strictly less than satisfying:(5) |

Theorem 1.4 may be compared with existing results in spectral graph theory on the “principal ratio” between the largest and smallest entries of the Perron vector of a connected graph. The known worst case lower bounds on this ratio are necessarily exponential in the diameter of the graph [CG07]. However, the worst case examples involve showing that two vertices separated by a long chain of degree 2 vertices have very different values in the Perron vector [TT15]. Theorem 1.4 articulates that there is always at least one vertex of non-maximal degree for which the ratio is only polynomial in the number of vertices.

The proof of Theorem 1.4 is based on an analysis of hitting times in the simple random walk on via electrical flows, and appears in Section 2. Combined with a perturbation-theoretic argument, it enables us to show that any small induced subgraph of the regular and connected graph , which must be irregular, can be extended to a slightly larger subgraph with significantly larger Perron value. This implies that closed walks cannot concentrate on small sets, yielding Theorem 1.3 in Section 3, which is finally used to deduce Theorem 1.2 in Section 4.

We show in Section 5.3 via an explicit example (Figure 2) that the exponent of appearing in Theorem 1.4 is sharp up to polylogarithmic factors.

###### Remark 1.5 (Higher Eigenvalues).

### 1.1 Related work

#### Eigenvalue Multiplicity.

Despite the straightforward nature of the question, relatively little is known about eigenvalue multiplicity of general graphs. As discussed in [JTY+19], if one assumes that is a bounded degree expander graph, then the bound of Theorem 1.1 can be improved to . On the other hand, if is assumed to be a Cayley graph of bounded doubling constant (indicating non-expansion), then [LM08] show that the multiplicity of the second eigenvalue is at most . In the context of Cayley graphs, one interesting new implication of Theorem 1.2 is that all Cayley graphs of degree have second eigenvalue multiplicity .

Distance regular graphs of diameter have exactly distinct eigenvalues (see [GOD93] 11.4.1 for a proof). However, besides the top eigenvalue (which must have multiplicity 1), generic bounds on the multiplicity of the other eigenvalues are not known. As expanding graphs have diameter , the average multiplicity of eigenvalues besides for expanding distance regular graphs is . It is tempting to see this as a hint that the multiplicity of the second eigenvalue could be .

Sublinear multiplicity does not necessarily hold for eigenvalues in the interior of the spectrum even assuming bounded degree. In particular, Rowlinson constructed regular graphs with an eigenvalue of multiplicity at least [ROW19].

#### Higher Order Cheeger Inequalities.

The results of [LRT+12, LGT14] imply that if a regular graph has a second eigenvalue multiplicity of , then its vertices can be partitioned into disjoint sets each having edge expansion . Combining this with the observation that a set cannot have expansion less than the reciprocal of its size shows that whenever for any , i.e., the graph is sufficiently non-expanding.

#### Support of Closed Walks.

There are as far as we are aware no known lower bounds for the support of a random closed walk of fixed length in a general graph (or even Cayley graph). It is relatively easy to derive such bounds for bounded degree graphs if the length of the walk is sufficiently larger than the mixing time of the simple random walk on the graph; the key feature of Theorem 1.3, which is needed for our application, is that the length of the walk can be taken much smaller.

#### Entries of the Perron Vector.

There is a large literature concerning the magnitude of the entries of the Perron eigenvector of a graph — see [STE14, Chapter 2] for a detailed discussion of results up to 2014. Rowlinson showed sufficient conditions on the Perron eigenvector for which changing the neighborhood of a vertex increases the spectral radius [ROW90]. Cvetković, Rowlinson, and Simić give a condition which, if satisfied, means a given edge swap increases the spectral radius [CRS93]. Cioabă showed that for a graph of maximum degree and diameter , [CIO07]. Cioabă, van Dam, Koolen, and Lee then showed that [CVK+10]. The results of [VSK+11] prove a lemma similar to Lemma 3.2, giving upper and lower bounds on the change in spectral radius from the deletion of edges. However, their result does not imply Lemma 3.2, and we prove a slightly different statement.

### 1.2 Notation

All logarithms are base unless noted otherwise.

#### Perron Eigenvector.

We use to denote the -normalized eigenvector corresponding to , which is a simple eigenvalue if is connected. Note that for connected , is strictly positive by the Perron-Frobenius theorem.

#### Electrical Flows.

#### Graphs.

For a graph , we denote by the subgraph induced by with adjacency matrix ; when is clear from the context, we will write instead of . A simple graph refers to a graph without multiedges or self-loops. An irregular graph is one which is not regular. We assume for all connected regular graphs, since otherwise the graph is just an edge, so .

## 2 Perron Eigenvector of Irregular Graphs

In this section we prove Theorem 1.4, which is a direct consequence of the following slightly more refined result.

###### Theorem 2.1 (Large Perron Entry).

Let be an irregular connected graph of maximum degree . Then there is a vertex with degree strictly less than satisfying:

(6) |

where .

###### Proof.

Write , where and . If then we are done, so assume not.
Let denote
the law of the simple random walk (SRW) on started at , and for any subset , let
denote the hitting time of the SRW to that subset; if is a singleton we will simply write .

Step 1. We begin by showing that there is a vertex adjacent to for which the random walk started at is reasonably likely to hit before (we will eventually choose to be a neighbor of this ). To do so, we use the well-known connection between hitting probabilities in random walks and electrical flows. Define a new graph by contracting all vertices in to a single vertex . Let be the vector of voltages in the electrical flow in with boundary conditions , regarding every edge as a unit resistor. By Ohm’s law, the current flow from to is equal to . We have the crude upper bound , so the outflow of current from is at least . By Kirchhoff’s current law, there must be a flow of at least on at least one edge . By Ohm’s law again, for this particular we must have

(7) |

where denotes the edge boundary of in . Appealing to e.g. [BOL13, Chapter IX, Theorem 8], this translates to the probabilistic bound

(8) |

Step 2. We now use (8) to show that is large. Let denote with a self-loop added to each vertex^{6}^{6}6This modification is only to ensure non-bipartiteness; if is not bipartite we may take and the proof works with in place of in (12)., and to ease notation let denote the law of the SRW on started at .
Note that and

(9) |

by (8) since adding self-loops to preserves electrical flows, which determine the probabilities in (9).

Since is connected and non-bipartite, the Perron-Frobenius theorem implies that:

for every . Writing for the Markov transition matrix of the SRW on and the diagonal matrix of degrees of vertices in , we find that

where is the degree of in .

We are interested in the ratio

(10) |

Fix an integer . The numerator of (10) is bounded as

(11) |

As the degree in of every vertex encountered before hitting is equal to , each conditional expectation appearing in (11) may be rewritten as

(12) |

since for all .

Observe that since is connected. Thus,

Combining this bound with (11) and (12), we have

Taking the limit as in (10) yields

Step 3. Since is adjacent to , we can choose a adjacent to . The eigenvector equation and nonnegativity of the Perron vector now imply , whence

as advertised. ∎

###### Remark 2.2.

As the proof shows, the right-hand side of (6) may be replaced with where is the set of vertices of degree strictly less than in and is the maximum effective resistance between two vertices in .

## 3 Support of Closed Walks

In this section we prove Theorem 1.3, which is an immediate consequence of the following slightly stronger result. Let denote the set of closed walks of length in rooted at with support at most .

###### Theorem 3.1 (Implies Theorem 1.3).

The proof requires a simple lemma lower bounding the increase in the Perron value of a subgraph upon adding a vertex in terms of the Perron vector.

###### Lemma 3.2 (Perturbation of ).

For any graph and vertex , the graph , which adds a vertex and the edge to , satisfies

###### Proof.

The largest eigenvalue of is at least the maximum of the quadratic form associated with of the unit vectors

for . We have and this quantity is maximized when

at which

∎

Combining Lemma 3.2 and Theorem 2.1 yields a bound on the increase of the top eigenvalue of an induced subgraph that may be achieved by adding vertices to it.

###### Lemma 3.3 (Support Extension).

For any connected -regular graph and any connected subset such that , there is a connected subset containing such that and

###### Proof.

Define and note that since contains at least one edge. As is connected, cannot be regular and has maximum degree . Therefore, by Theorem 2.1, there is some vertex with and

where . As is a normalized vector with entries, . Therefore . Take to be any vertex in that neighbors in . By Lemma 3.2,

(15) |

Assuming that , we can iterate this process times, adding the vertices . At each step we add the vertex and increase the spectral radius of by at least . Therefore, defining , we have

where the last inequality follows from approximating the sum with the integral. As , this translates to the desired multiplicative bound. ∎

###### Proof of Theorem 3.1.

We begin by showing (13). Let be the set of connected subgraphs of with vertices containing . Choose to be the maximizer of among , and let be the extension of guaranteed by Lemma 3.3 to satisfy

Notice that

since each walk is contained in at least one . Furthermore, since each subgraph of may be encoded by one of its spanning trees, which may in turn be encoded by a closed walk rooted at traversing the edges of the tree. It follows that

(16) |

We claim that for every ,

(17) |

To see this, let be a path in of length between and , which must exist since is connected and has size . Then every closed walk of length in rooted at may be extended to a walk of length in rooted at by attaching and its reverse. Since all of the walks produced this way are distinct and , inequality (17) follows.

Choose to be the maximizer of , for which we have:

Combining this with (17) and substituting in (16), we obtain:

Applying the inequality for and the bound , we obtain

(18) |

which implies

whenever

establishing (13).

We now show (14) via a small modification of the above proof. Assume . The key observation is that each vertex has at least edges in to other vertices, so in a subgraph of size at most every vertex has at least one edge in leaving the subgraph. In this case, we can simply choose as in Lemma 3.3. Therefore (15) can be improved to

Therefore, after adding vertices to according to the process of Lemma 3.3, we find a set satisfying

## 4 Bound on Eigenvalue Multiplicity

In this section we prove Theorem 1.2, restated below in slightly more detail.

###### Theorem 4.1 (Detailed Theorem 1.2).

###### Proof.

For now, assume that . Let denote the set of all closed walks in of length , denote such walks with support at most , and .

Set and and let be a parameter satisfying

(21) |

to be chosen later. Delete vertices from uniformly at random. If , the probability that none of the vertices of are deleted is at most

It then follows by the probabilistic method that there exists a deletion such that the resulting subgraph of contains at most walks from .

Write and let be the number of eigenvalues of in the interval for . Since is even,

We may assume that the diameter of is at least since otherwise , making the theorem statement vacuous. Then the Alon-Boppana bound [NIL91] states that implying . Note that for sufficiently large ,

since we have assumed . Thus, . Combining these facts,

Comments

There are no comments yet.