A strengthening and a multipartite generalization of the Alon-Boppana-Serre Theorem
The Alon-Boppana theorem confirms that for every and every integer , there are only finitely many -regular graphs whose second largest eigenvalue is at most . Serre gave a strengthening showing that a positive proportion of eigenvalues of any -regular graph must be bigger than . We provide a multipartite version of this result. Our proofs are elementary and work also in the case when graphs are not regular. In the simplest, monopartite case, our result extends the Alon-Boppana-Serre result to non-regular graphs of minimum degree and bounded maximum degree. The two-partite result shows that for every and any positive integers , every -vertex graph of maximum degree at most , whose vertex set is the union of (not necessarily disjoint) subsets , such that every vertex in has at least neighbors in for , has eigenvalues that are larger than . Finally, we strengthen the Alon-Boppana-Serre theorem by showing that the lower bound can be replaced by for some if graphs have bounded “global girth”. On the other side of the spectrum, if the odd girth is large, then we get an Alon-Boppana-Serre type theorem for the negative eigenvalues as well.
Keywords: spectral radius, eigenvalue, Ramanujan graph, universal cover.
Math. Subj. Classification: 05C50
After the breakthrough paper by Alon and Milman  in 1985, it became apparent that regular graphs, whose spectral gap (i.e. the difference between the largest and the second largest eigenvalue) is large, posses some extraordinary properties, like unusually fast expansion and resemblance to random graphs. This led to the definition of Ramanujan graphs. These are -regular graphs whose second largest eigenvalue does not exceed the value . Lubotzky, Phillips, and Sarnak , and independently Margulis , were the first to show that Ramanujan graphs exist. Their constructions are based on number theory and work when the degree is equal to for some prime . Later, several new constructions were discovered, showing that Ramanujan graphs exist for every degree which is of the form for some integer and some prime , see Morgenstern .
It is not immediately clear why the special choice of is taken when defining Ramanujan graphs. One reason is that this is the spectral radius of the infinite -regular tree, which is the universal cover for all -regular graphs. Another reason is the following result of Alon and Boppana (see ) which shows that this is the smallest number that makes sense.
Theorem 1.1 (Alon-Boppana)
For every and every , there are only finitely many -regular graphs whose second largest eigenvalue is at most .
Alternative proofs of Theorem 1.1 were given by Friedman  and by Nilli , who has recently further simplified her arguments in . Actually, the proofs in [8, 21] imply a stronger version of Theorem 1.1 by making the same conclusion for more eigenvalues than just the second largest one. This strengthening is attributed to Serre  (see also [5, 7, 12]), who stated the following quantitative version of the Alon-Boppana Theorem:
Theorem 1.2 (Serre)
For every positive integer and every , there exists a constant such that every -regular graph of order has at least eigenvalues that are larger than .
In this paper we give a multipartite generalization of the Alon-Boppana Theorem, see Theorem 5.1. The Ramanujan value is replaced by the spectral radius of the universal covering tree of the multipartite parameters (cf. Section 2 for definitions). Our proof has similarities with Nilli’s proof , and seems to be even simpler if restricted to the special case of -regular graphs. The main step is based on the interlacing theorem and is entirely elementary.
Our proofs work also in the case when graphs are not regular. In the simplest, monopartite case, our result extends the Alon-Boppana-Serre result to non-regular graphs of minimum degree and bounded maximum degree. A strengthening of this form has been obtained previously by Hoory . In the next simplest two-partite case it is shown that for every and any positive integers , every -vertex graph of maximum degree at most , whose vertex set is the union of (not necessarily disjoint) subsets , such that every vertex in has at least neighbors in for , has eigenvalues that are larger than .
After submission of this paper, S. Cioabǎ informed us about some related work. Greenberg  obtained a generalized version of the Serre theorem in a similar form as ours, but only claiming that there are eigenvalues whose absolute value is larger than . Cioabǎ  improved Greenberg’s work to the same form as given in Theorem 1.2. Greenberg’s result also appears in .
In the last section we tailor the proofs to obtain a strengthening of the Alon-Boppana-Serre theorem by showing that the lower bound can be replaced by for some if graphs have bounded universal girth (see Section 6 for the definition). On the other side of the spectrum, if the odd girth (i.e. the length of a shortest odd cycle) is large, then we obtain an Alon-Boppana-Serre type theorem for the negative eigenvalues.
If is a (finite) graph, we denote by the th largest eigenvalue of the adjacency matrix of , respecting multiplicities. The largest eigenvalue of , , is also referred to as the spectral radius of . It follows from the Perron-Frobenius theorem (see, e.g. ) that is an eigenvalue of that has an eigenvector whose coordinates are all non-negative. Moreover, if is connected, then is strictly positive.
If is an integer, a set of vertices of a graph is said to be -apart if any two vertices in are at distance at least in . We denote by the maximum cardinality of a vertex set in that is -apart. Note that is the usual independence number of the graph.
Let be a graph, , and let be an integer. We denote by the induced subgraph of on vertices that are at distance at most from . The subgraph is called the -ball around in .
We allow infinite graphs, but they will always be locally finite. In particular, the -ball around any vertex of a graph is always finite.
2 Universal covers and subcovers
Let be a square matrix of order , whose entries are non-negative integers. For , we define the degree in as the integer . Suppose that further satisfies the following conditions:
If , then also .
The graph of is connected, i.e., for every there are integers in , where , , and for .
For every sequence of (distinct) integers in , we have
Such a matrix is called a -partite degree matrix.
Let be a -partite degree matrix. If a graph admits a partition of its vertex set into classes, , such that every vertex in has precisely neighbors in , for all , then we say that is a -partite degree matrix for . The corresponding partition is said to be an equitable partition for ; see, e.g. .
Let be a -partite degree matrix.
(a) There exists a finite graph whose degree matrix is .
(b) There exists a tree whose degree matrix is . The tree is determined up to isomorphism.
Proof. (a) First we remark that the condition (D3) implies that the set of equalities , , has a positive solution . Since all are integers, there is a solution whose values () are positive integers. To obtain a graph , we take vertex sets of cardinalities for , and join and so that the edges between them form a -biregular bipartite graph. Then it is clear that is a -partite degree matrix for .
(b) To get , we just take what is known as the universal cover of the graph obtained in part (a).
We add a short proof of existence of that does not use the property (D3) which is needed in (a). Let us first assume that for . This case is intuitively clear and we leave the details of the proof for the reader. Note that is always infinite in this case.
The rest of the proof is by induction on . We may assume that . If , then is either a single vertex (if ) or an edge (if ). If , then (D2) implies that and the non-zero element in row of is not . Thus, there is a unique such that . Let . By (D1), we conclude that and all other elements in the column of are zero. Let be the submatrix of obtained by deleting the last row and the last column. Since this operation acts like removing a vertex of degree 1 from a graph, still satisfies (D1)–(D2), and hence we can apply the induction hypothesis to find the tree . Finally, we obtain by adding, to each vertex in , pendant edges. All new vertices are of degree 1 and form the class in .
The tree is called the universal cover of the multipartite degree matrix . Let be the corresponding equitable partition of . If is any graph whose -partite degree matrix is , there is a covering projection which maps vertices in onto the th class of the equitable partition of .
Covering projections, universal covers and equitable partitions are regularly used in algebraic graph theory. In the sequel we shall introduce a weaker notion, distinguished by the prefix ‘sub’, in which only those properties that are important for our main results will be preserved.
A degree matrix is said to be a -partite subdegree matrix for a graph if there is a graph homomorphism which is locally -, i.e., for each vertex , maps edges incident with injectively to the edges incident with . The homomorphism is called a subuniversal projection and the tree is a subuniversal cover of .
If is a subuniversal projection, let , . Then it is easy to see that the (not necessarily disjoint) vertex-sets satisfy the following condition: Every vertex in has at least neighbors in , for all . This gives a necessary condition for existence of a subuniversal projection. Unfortunately, this condition is not sufficient. But if we ask that every vertex in has at least neighbors in , for all , then the existence of a subuniversal projection to is easily verified.
Suppose that is a subdegree matrix for a (possibly infinite) graph , and let be the corresponding subuniversal cover. Let and let be a vertex that is mapped to via a subuniversal projection . Then for every , the spectral radius of the -ball in is at least as large as the spectral radius of the corresponding -ball in , .
Proof. The spectral radius of a connected graph can be expressed as
where denotes the number of closed walks of length in starting at the vertex . Every closed walk in starting at is projected by to a closed walk in starting at . The projection of these walks is 1-1, since is locally 1-1. Hence,
This inequality in combination with (1) implies that .
3 The spectral radius of infinite trees
If is a connected infinite (locally finite) graph, we define its spectral radius as
where is any vertex of . It is easy to see that the limit exists (it may be infinite if the degrees of have no finite upper bound) and that it is independent of the choice of . The spectral radius of infinite graphs defined above coincides with the notion obtained through the spectral theory of linear operators in Hilbert spaces; we refer to  for an overview.
The monotonicity property of the spectral radius of finite graphs implies that for every connected finite graph and any proper subgraph of , we have . Since is connected, infinite, and locally finite, for every . This implies that
Let us remark that the spectral radius of an infinite -regular tree is equal to , the value that appears in the definition of Ramanujan graphs. This was proved by Kesten , see also Dynkin and Malyutov , Cartier , and Woess . We will use the spectral radius of universal cover trees introduced in the previous section to replace the Ramanujan bound with the corresponding bound suitable for our multipartite generalization.
In the special case when the graph is the infinite -regular tree, which shall be denoted by , it is easy to determine the precise rate of convergence in (3).
For every integer , we have
Proof. Let denote the number of closed walks of length . It will be convenient to consider the subtree of which is equal to the connected component containing the vertex of the subgraph obtained after deleting an edge of incident with . The vertex has degree in , while all other vertices still have degree . The tree has a natural projection onto the one-way-infinite path (whose vertices we denote by the non-negative integers ) such that all vertices at distance from are mapped onto the vertex in . Every closed walk (based at ) of length in is projected onto a closed walk in based at the vertex . Moreover, the -ball in is projected onto the path on vertices .
Whenever we walk away from in , we have choices to do so. This implies that
When , the quantities raised to the power tend to the spectral radii of the corresponding graphs, and we conclude that . Since is a proper finite subgraph of , this implies the (strict) inequality of the theorem.
The rate of convergence is likely the same for more general universal covers of finite graphs. We propose the following conjecture.
For every multipartite degree matrix , there exists a constant such that for every , we have
4 Multipartite Ramanujan graphs
In this section we introduce a generalized notion of Ramanujan graphs. The following lemma shows that we cannot simply compare with as is the case for -regular graphs.
If is a -partite degree matrix, then all eigenvalues of are real and their algebraic multiplicity is equal to their geometric multiplicity. If is a multipartite degree matrix of a finite graph , then every eigenvalue of is also an eigenvalue of . Moreover, .
Proof. Let be a positive solution of the system , , which was shown to exist in the proof of Lemma 2.1(a). If is the diagonal matrix of order whose entry is equal to (), then is a symmetric matrix. This implies the first part of the lemma.
To verify the second part, let be an eigenvalue of , and let be an eigenvector for . Let be the partition of corresponding to the degree matrix . If we set for every , then it is easy to see that is an eigenvector of the adjacency matrix of for the eigenvalue .
To prove the last claim, observe that the eigenvalue has a positive eigenvector by the Perron-Frobenius Theorem. Its lift in is a positive eigenvector of for the eigenvalue . Again, by applying the Perron-Frobenius Theorem, we conclude that this eigenvector corresponds to the largest eigenvalue of .
Let be a -partite degree matrix. Let be the largest integer such that . Note that exists since . We say that a finite graph with degree matrix (resp. subdegree matrix) is -Ramanujan (resp. -Ramanujan) if . We believe that there is an abundance of generalized Ramanujan graphs and propose the following conjectures (in which we assume that the minimum degree of is at least 2).
If there exists a -Ramanujan graph for a multipartite degree matrix , then there exist infinitely many -Ramanujan graphs.
If there exists a -Ramanujan graph for a multipartite degree matrix , then there exist infinitely many -Ramanujan graphs.
If is a degree matrix of order , and , then there exist infinitely many -Ramanujan graphs.
One cannot exclude the possibility that there exist infinitely many -Ramanujan graphs for every degree matrix , but our knowledge is too limited at this point to propose this as a conjecture.
5 A generalized Alon-Boppana-Serre theorem
Let be a multipartite degree matrix, and let .
(a) For every , there exists an integer such that for every integer and for every graph , if is a subdegree matrix of and , then .
(b) For and every positive integer , there exists a constant such that every graph of order , of maximum degree at most and with subdegree matrix , has at least eigenvalues that are larger than .
Proof. (a) Let be the smallest integer such that , and let and be as specified. Since , there are vertices that are -apart. The -balls around these vertices are not only pairwise disjoint, but also form an induced subgraph of . By the eigenvalue interlacing property for induced subgraphs, we know that
By Theorem 2.2 and by our choice of , we have
This completes the proof of (a).
(b) This part follows from (a). It is just to be noted that any -ball in contains at most vertices. Thus, , and hence part (a) applies with .
It is worth mentioning that the condition involving in Theorem 5.1(a) is necessary if we only assume that is a subdegree matrix. Simple examples showing this are provided by the family of all complete graphs whose second largest eigenvalue is always equal to , or by the family of all complete bipartite graphs whose second eigenvalue is 0.
will do the job.
As an example, let us consider the following special case. The bipartite degree matrix
involves, in particular, all bipartite graphs with bipartition , whose degrees in are at least and whose degrees in are at least . The spectral radius of is (cf. )
Let be positive integers, and let be the set of all graphs whose maximum vertex degree is at most and whose vertex set is the union of (not necessarily disjoint) subsets , such that every vertex in has at least neighbors in for . For every , every -vertex graph has eigenvalues larger than .
Proof. We claim that there exists a subuniversal projection , where is the degree matrix given in (5). The tree is -biregular. We map a vertex of degree in onto the vertex . After fixing , we extend the mapping to a locally 1-1 homomorphism in a greedy fashion (by taking the breadth-first search order of vertices of starting at ) so that vertices of degree are mapped to , .
Let be a neighbor of in and let . Theorem 2.2 shows that for large enough ,
Since the maximum degree of is bounded by , the -balls in have bounded number of vertices, say at most . Therefore, , and so there are at least this many pairwise non-adjacent induced -balls around vertices in . As before, the eigenvalue interlacing theorem and (6)–(7) imply that linearly many eigenvalues of are larger than
6 Global girth and Ramanujan graphs
All known Ramanujan graphs are Cayley graphs and their girth increases with their order. We shall use the method of this paper to explain why the girth cannot be bounded. Actually, we shall prove that a small girth condition implies that -regular graphs are “far from being Ramanujan;” see Theorem 6.2 below.
Let be a graph. A closed walk is retracting-free if for (where and ). It is easy to see that if is a finite graph with minimum degree at least 2, then for every vertex of there exists a retracting-free closed walk through .
Let be the length of a shortest retracting-free closed walk through . The universal girth of , denoted by , is the smallest integer such that every vertex in has a retracting-free closed walk of length . Let us observe that is at most the least common multiple of the values , . Also, if is vertex-transitive, then is equal to the girth of .
Let be the graph obtained from the -regular tree by expanding each vertex into the cycle of length , such that each vertex of is incident with of the edges of incident with . See Figure 1 showing the case of and . The graph is the Cayley graph of the free product of copies of and one copy of (with the natural generating set).
Paschke  determined the spectral radius of :
Theorem 6.1 (Paschke)
For and , the graph has spectral radius
Paschke  used this result to provide a non-trivial lower bound on the spectral radius of infinite vertex-transitive graphs of the given girth . He showed that a vertex transitive -regular graph containing a -cycle has spectral radius at least ). The formula in Theorem 6.1 gives a lower bound of the form
where is a function such that such that for every , , and for every , .
Now, we strengthen the Alon-Boppana-Serre theorem by showing that the lower bound can be replaced by for some if graphs have bounded universal girth.
For every and every , there exist and such that every -vertex graph with minimum degree at least , maximum degree at most and universal girth at most has at least eigenvalues that are larger than .
Proof. The proof follows the same pattern as the proof of Theorem 5.1, except that we use the graph , where is the universal girth of , playing the role of the universal cover . Here, we have to take large enough so that . Such an exists because of (3) and since .
It is straightforward to generalize the proof of Theorem 6.2 to the setting of degree matrices. What we need is just an analogue of the Paschke theorem. However, we do not intend to dig into the details in this note.
7 The other side of the spectrum
As shown in the previous section, small universal girth yields improved lower bounds on large eigenvalues, so Ramanujan graphs must have growing girth. On the other hand, large girth has some further cosequences. In particular, it shows that the negative eigenvalues satisfy the Alon-Boppana-Serre property as well.
Let us first formulate the monopartite version for the negative eigenvalues. It involves the notion of the odd girth of the graph, meaning the length of a shortest cycle of odd length in the graph. (If is bipartite, then the odd girth is .) This result was obtained earlier by Friedman  and Nilli ; it also appears in Ciaobǎ  (with a slightly weaker estimate of ).
For every and , there exists a positive constant such that every graph of order , of minimum degree , maximum degree at most , and with odd girth at least has at least eigenvalues that are larger than and has at least eigenvalues that are smaller than , where .
Proof. (Sketch) The proof is essentially the same as the proof of Theorem 5.1(b), where we take and apply the estimate of Theorem 3.1. The assumption that the odd girth is more than shows that the -balls in contain no cycles of odd length. In particular, they are bipartite and hence their spectrum is symmetric with respect to 0. Thus, knowing that the spectral radius is large, we conclude that the smallest eigenvalue is large in absolute value. Now, we can use the interlacing theorem for the smallest eigenvalues of compared to the eigenvalues of the induced subgraph of consisting of disjoint -balls around vertices that are -apart.
The generalized version of Theorem 7.1 holds as well. The proof is the same, except that we do not provide an explicit estimate on in terms of the odd girth.
Let be a multipartite degree matrix, and let . For every and every positive integer , there exists an integer and a positive constant such that every graph of order , of maximum degree at most , with subdegree matrix , and with odd girth at least has at least eigenvalues that are smaller than .
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