On the Spectrum of Finite, Rooted Homogeneous Trees

03/17/2019 ∙ by Daryl R. DeFord, et al. ∙ MIT 0

In this paper we study the adjacency spectrum of families of finite rooted trees with regular branching properties. In particular, we show that in the case of constant branching, the eigenvalues are realized as the roots of a family of generalized Fibonacci polynomials and produce a limiting distribution for the eigenvalues as the tree depth goes to infinity. We indicate how these results can be extended to periodic branching patterns and also provide a generalization to higher order simplicial complexes.

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1. Introduction

For integers , let denote the -ary rooted tree of depth . That is, is first of all a tree (in the graph-theoretic sense – so a graph without cycles) with a distinguished node, the “root” that orients all the nodes in in the sense that any node is located at some depth defined as the distance of that node from the root. As usual, the leaves of are the nodes most distant from the root and have degree . The tree is of depth , if the leaves are at distance from the root. The rooted tree is -ary in the sense that every non-leaf node has children defined as the nodes connected to it that are at depth one more than the node itself. Thus, an interior node is also a parent of children. A node at distance from the root is of generation . Note that in the root has degree , interior nodes have degree and leaves have degree . The well-known binary tree of depth is .

This kind of regular branching – constant number of children at every non-leaf node – has natural extensions. For example, one can consider more general periodic branchings: given a vector

, define the rooted tree to be the tree that has successions of branchings with children of the root, children for the nodes at level and in general, a node of generation where and has children.

In this paper we consider properties of the spectrum of the trees , focusing mainly on the simple case of , but where possible, extending to vectors of longer length and finite subtrees of well–known infinite trees. By “spectrum” we mean the spectrum of the adjacency operator of the tree (graph). These trees can be viewed as simplicial complexes of dimension with some structural regularities. This view admits at least one natural two-dimensional generalization (with corresponding adjacency operator) and we also include some preliminary investigations of this higher order context.

1.1. Related Work

Homogeneous trees with constant branching have found application in a wide variety of fields. In particular, finite versions of these trees where all non-leaf vertices have the same number of edges have found frequent application in computer science. The infinite case is also well studied. The vector-valued form have been studied extensively in the context of dynamical systems – wherein their consideration arises in a naturally in the context of the dynamics of infinitely renormalizable maps on the unit interval and in the considerations of periodic orbit structures for maps on the unit interval [1]. The -regular infinite tree is a homogeneous space for the -adic Lie group wherein the associated Hecke operator takes the place of the adjacency matrix, well studied for various number theoretic connections [3].

Our results are also related to many recent results on the expansion properties of graphs. See [8] for a survey and examples of how methods from trees can be extended to arbitrary regular graphs. These methods are mostly concerned with deriving bounds on the spectral gap for regular graphs instead of computing the entire spectrum explicitly. Additionally, there has been recent work on similar problems for higher order simplices [10].

(a)
(b)
Figure 1. The two principal types of rooted homogeneous trees considered in this paper. Note that each non–leaf has children while in the subtrees of the infinite regular tree each non–leaf has degree .

1.2. Main Results

Our main results are for the rooted trees For the rooted tree naturally embeds into any rooted tree

by identifying the roots. We will see that the spectra of these trees also have a natural nesting and moreover, use the nested structure to relate the associated eigenvectors as well.

More specifically, we are able to characterize the eigenvalues as roots of families of polynomials derived from the recursive structure of the trees. To be more precise, for , define polynomials via the recurrence

(1.1)

with initial conditions

(1.2)

Thus, for , is of degree .

Theorem 1.

Let and . The the roots of are precisely the eigenvalues of the -ary tree .

Corollary 1.

If then the spectrum of is a subset of the spectrum of .

Theorem 2.

If is a root and not a root of for any then asymptotically (as ), the proportion of eigenvalues of equal to is .

2. Spectrum of

2.1. Preliminaries.

Let denote the -dimensional vector space of complex-valued functions on . It is not difficult to derive that

(2.1)

The standard basis for is the set of functions indexed by the nodes of that are equal to at the given node and elsewhere. Thus any can be written

(2.2)

so that is thus identified with the values that it takes on the nodes of . This is the natural representation of for considering the effect of the adjacency operator

(2.3)

where denotes that is adjacent (linked to) .

We call isotropic (with respect to the root) if is constant over all at a given distance from the root. In that case, we let denote the common value that takes for at distance from the root. It is easy to see that takes isotropic functions to isotropic functions.

The spectrum of will be characterized by the zeros of the polynomials . These are exactly the generalized Fibonacci polynomials of type defined in [7]. These polynomials were introduced in [11] and many of their general properties were reported in [6, 7]. In Lemma 1 we record some of their properties that we will use throughout the paper.

Lemma 1.

The polynomials have the following properties:

  1. If is a root of and then is not a root of .

  2. The polynomials form a divisibility sequence, that is for , then .

  3. The roots of are exactly .

Proof.

These results are Theorems 4, 6, and 10 in [7]. ∎

2.2. Eigenvalues for and roots of .

We first show the “forward” direction of Theorem 1: if is a root of for then is an eigenvalue of . We do this by explicitly constructing eigenvectors for these using the standard basis.

Lemma 2.

Let be isotropic with for a root of . Then is an eigenvector for with eigenvalue .

Proof.

Let be defined as in the statement of the lemma. We now confirm the eigenrelation for . We will make repeated use of the defining properties of the polynomials (see Eqns. (1.1,1.2)).

Case 1: . In this case we are at the leaves. Thus

Case 2: . In this case we are at a node that is neither a leaf nor the root. Then

where the second-to-last line uses the defining recurrence of the .

Case 3: . In this case, we are at the root and

where the second-to-last line uses the definition of the and the fact that is a root of .

Note that the only case of the proof that depended on the choice of as a root of was Case 3. We now show how this eigenvector construction can be extended to the roots of the other . In short, for any distance from the root, the construction can be modified by constructing vectors that are linear combinations of analogous functions, isotropic with respect to rooted subtrees issuing from a node at distance . We’ll do it in pieces.

Lemma 3.

If is a root of then is an eigenvalue of . Moreover, the dimension of the

-eigenspace is

.

Proof.

As in the previous lemma we construct eigenvectors directly. We begin by selecting real numbers such that and not all the are zero. Note that this is a -dimensional subspace. Order the children of the root arbitrarily from to and for each descendant of child at distance from the root, set the corresponding eigenvector value to . Finally, set the value at the root to .

In this case, the eigenrelation at the root is satisfied since the sum of the values assigned to the children of the root is . Note that while is not isotropic on , it is isotropic on any subtree rooted at any child of the root and extending through all of its descendants. The argument in the previous lemma is easily adapted: the only change is that each of the relations for the various cases are multiplied by except possibly for the case of the root’s children themselves – the new “root” of the subtree. For this, consider any given child of the root :

where the second line follows from (1.1) and the third line follows from the choice of and the definition of .

The construction of Lemma 3 can be generalized: instead of only prescribing the root to be zero, we can now construct eigenfunctions that are zero out to some fixed distance from the root and then mimic the construction. We say an eigenvector

, with associated eigenvalue , of is of type if in its standard basis representation it has at least one nonzero coefficient for a node (index) at distance from the leaves and no nonzero entries at distance greater than from the leaves. As an example, the eigenvectors constructed in Lemma 2 are of type for a root of .

Lemma 4.

If is a root of for then there is a dimensional eigenspace for consisting of eigenvectors with type for .

Proof.

To construct these eigenvectors we begin by setting the coefficient at the root and the coefficients corresponding to all nodes whose distance to the root is less than or equal to to zero. Recall that there are precisely nodes at distance exactly from the root. For each node at distance from the root, we select complex numbers so that .

Following the construction given in Lemma 2 we order the children of each of these nodes arbitrarily and for each descendant of node at distance we set the corresponding eigenvector coefficient to . By the arguments given in Lemma 1 and Lemma 2 these vectors satisfy the eigenvector equation for the adjacency operator. Since we have degrees of freedom at each of the nodes this proof is complete.

Theorem 3.

The roots of for are precisely the eigenvalues of the finite -ary tree.

Proof.

It suffices to show that we have constructed linearly independent eigenvectors corresponding to these values. Summing up over the eigenvectors of type from Lemma 4, recalling that is a degree polynomial gives eigenvalues. This sum telescopes, leaving eigenvalues accounted for. Adding the remaining eigenvalues from completes the proof.

2.3. Limiting Distribution

In this section, we explore the behavior of the distribution of the eigenvalues as . As we saw above, the multiplicities of the eigenvalues that correspond to roots of continue to grow exponentially with . Qualitatively, a (suitably normalized) plot for large of the eigenvalues of with multiplicities should look like a collection of horizontal line segments. We begin by determining how the number of unique values grows as .

Lemma 5.

The number of roots of that are not roots of is

Proof.

Note that is a polynomial of degree . Since is a divisibility sequence, if is prime then is irreducible and has roots. We proceed by strong induction on . Since and are prime and have roots and respectively our base case is complete. For the inductive step consider the number of new roots of . There are total roots and each divisor of contributes by the inductive hypothesis:

as desired.

Next, we determine the number of occurrences of each of the new values from the previous lemma. The reason that this question is not answered by Lemma 4 is that since is a divisibility sequence, the values continue to reoccur. For determining the total number of eigenvalues we did not have to consider the effects of these repetitions, as the corresponding eigenvectors as in Theorem 4 are linearly independent. However, now we are interested in counting the total multiplicities of the values themselves.

Lemma 6.

Let be a root of that occurs for the first time and let be the function that returns 1 if and 0 otherwise. Then, for the eigenvalue occurs with multiplicity

Equivalently, for large the proportion of eigenvalues of of value converges to .

Proof.

Let be a root of that occurs for the first time. By the divisibility property we know that is a root of for all . From Lemma 4 this gives that the total multiplicity of as an eigenvalue for is .

Rewriting this as a geometric series allows us to simplify:

Since the total number of nodes in is this gives the asymptotic proportion as . ∎

These results have an interesting corollary, a version of which has been previously proved using Lambert Series:

Corollary 2.

When (the binary tree) this gives the following simple sum:

Corollary 3.

The previous propositions are explain the characteristic “Devil’s Staircase” shape of ordered plot of eigenvalues of . That is, for each of the values that are roots of occurring for the first time, the width of the “stair” corresponding to each in the plot is proportional to .

2.4. Singular Distributions

Another way to interpret the results of this section is to consider the sequence of uniform distributions over the eigenvalues, with multiplicities, of

as . The condition in Definition 1 gives that the limit of these distributions is a singular distribution, like the Cantor function. In order to formalize this discussion it is convenient to normalize the eigenvalues of and consider the set:

Then the sequence of uniform distributions over converges to a singular distribution as . This raises the interesting question of the properties of the measure zero support set for the limiting distribution. Lemmas 6 and 7 allow us to write expressions for these endpoints.

Theorem 4.

The endpoints of the support of are:

Proof.

We proceed by directly computing the endpoints of each interval corresponding to a new root of . Lemma 7 gives that the width of the corresponding interval is , so it suffices to compute the left endpoint. By part (iii) of Lemma 1 we know that these values are of the form where . Since is monotonic on and for all the intervals that occur to the left of any given value are exactly those for which .

Thus, summing the widths over all possible values of the left endpoint of the interval is given by:

and the right endpoints follow by adding the width of the relevant interval. ∎

(a)
(b)
Figure 2. Normalized eigenvalues of adjacency spectra. Notice the transient eigenvalues associated to the polynomials in plot (B) for the -regular (except for the leaves) version .

3. Finite Subtrees of Infinite Regular Trees

Another natural set of trees to consider is the collection of regular rooted finite subtrees of the infinite regular trees. The difference between these and those considered above is that the trees described in this case the root has children but each other non–leaf node in these trees has children so that each non–leaf has exactly neighbors. These are the regular finite subtrees that are encountered in the context of buildings and -adic homogeneous spaces [2]. We will denote these trees by .

In order to represent the eigenvalues of these trees in terms of polynomials as in Section 2 we need to define another family of polynomials in order to satisfy the eigenvector relation at the root. For , let be a polynomial of degree satisfying the recurrence

(3.1)

with initial conditions and .

The main results of the previous setting can be translated to these trees with minimal adjustments to the proofs.

Lemma 7.

Let be depth regular with for a root of . Then is an eigenvector for with eigenvalue .

Lemma 8.

If is a root of then is an eigenvalue of . Moreover, the dimension of the -eigenspace is .

Lemma 9.

If is a root of for then there is a dimensional eigenspace for consisting of eigenvectors with type for .

Lemma 10.

The dimension of the eigenspaces of corresponding to eigenvalues that are roots of for are equal to:

Proof.

Let be the sum of the multiplicities of the eigenvalues of that correspond to roots of for . We claim that satisfies the recurrence relation . We can divide the eigenvalues into three classes: those that occur with multiplicity at least in , those that occur with multiplicity exactly in , and those that do not occur in .

For the eigenvalues in case 1, Lemma 3 tells us that each value occurs with times the multiplicity in . For the eigenvalues in case 2, Lemma 3 tells us that overcounts by a factor of . Finally, for case 3, Lemma 2 gives us exactly new linearly independent eigenvectors. Putting these cases together we get .

This is a recurrence relation with eigenvalues so it can be represented as a generalized power sum [5] of the form . Using the initial conditions , , and we can compute , , and which gives as desired. ∎

Theorem 5.

The roots of for and are precisely the eigenvalues of the finite rooted subtrees of the -ary tree.

Proof.

Lemma 10 provides eigenvalues of . Since is a degree polynomial, the construction in Lemma 1 gives us the remaining eigenvectors. ∎

3.1. Limiting Distribution

As in the cases discussed in Section 2 the reoccurrence of the eigenvalues in the divisibility sequence means that the limiting distribution of the eigenvalues for these trees can be computed. Interestingly, the limit identities of Corollaries 2 and 3 are also expressed in these trees with the change of variable for . Although this is a little unintuitive due to the addition of the eigenvalues, note that the size of the tree grows exponentially in while the degree of grows linearly. Thus, asymptotically, almost all of the eigenvalues of trees of sufficiently large depth are roots of for some .

Lemma 11.

Let be a root of that occurs for the first time. Then, for the eigenvalue occurs with multiplicity at least111It is possible for to occur as both a root of and . For example, there are –eigenvalues of both types in .

in . Equivalently, for large the proportion of eigenvalues of of value is approximately .

Proof.

Let be a root of that occurs for the first time. By the divisibility property we know that is a root of for all . From Lemma 4 this gives that the total multiplicity of as an eigenvalue for is at least .

Rewriting this as a geometric series allows us to simplify:

Since the total number of nodes in is this gives the asymptotic proportion as . ∎

Theorem 6.

The endpoints of the support of are:

3.2. Eigenvalues on Infinite Trees

The results from earlier in this section can be used to give explicit constructions of eigenvectors for the Hecke operators on the infinite tree that are stable under congruence subgroups of .

Theorem 7.

Let be an eigenvalue of with corresponding eigenvector . Then, can be extended to a unique eigenvector of stable under .

Proof.

We construct the associated Hecke eigenvector, , directly by computing the values on the nodes at each distance from the root. We begin by assigning the values from to for the corresponding nodes in the infinite tree . As already satisfies the eigenvalue equation for we must assign the value to all nodes at depth in the infinite tree.

Continuing on, the stability condition enforces that all nodes with the same immediate parent must be assigned to the same value. Thus, for each each leaf of the values assigned to the nodes below the corresponding node at depth in the infinite tree must satisfy with initial conditions given by the value of the leaf in and . This recurrence completely determines and we have constructed the desired Hecke eigenvector.

4. Periodic Branching

We now turn to trees with non–constant branching behavior. We will see that the spectral behavior of these trees with periodic branching is quite similar to the homogeneous trees discussed earlier. In order to consider sequences of trees that are spectrally stable, we derive the adjacency spectra for trees with complete periods of branching, (e.g., so that the motif is not interrupted).

We will carefully derive the construction for 2–periodic trees (like one with branching) and then show how the method can be generalized to longer periods.

4.1. trees

In this section we construct the eigenvalues and eigenvectors of trees with branching pattern of finite length. As in the case of homogeneous trees we will construct the eigenvalues as roots of a sequence of polynomials.

We proceed as in Section 2 defining two families of polynomials whose roots are the eigenvalues of the trees. However, the branching process makes determining the relationships between the polynomials at each layer more complex (Figure 7). In order to derive a relation for the polynomials we define a system of recurrences:

where represents the nodes with children and the represent nodes with children. In order to derive a recurrence relation for the and the separately we use the determinant operator method of [5] Theorem 7.20. Letting denote the sequence shift operator, our system can be summarized:

The determinant of this matrix is so the sequences individually satisfy the recurrence: and this determines the recurrence that defines our polynomials . Thus, we define with initial conditions , , , and and define . Note that applied to the tree this becomes a fourth order recurrence that only involves even indexed terms as the and sequences are independent.

Theorem 8.

has as an eigenvalue if and only if is a root of for or a root of .

Proof.

We follow the outline Lemmas 2–5. As in Section 2, it is straightforward to construct eigenvectors of that are roots of for or a root of . The one subtlety to note is that since we are growing the tree two layers at a time there are two sets of roots for the , those for that occur with non–zero values on the direct children of the root and those of that occur with zero values on the direct children of the root as well.

In order to show that these are all of the eigenvalues it again suffices to show that we have constructed linearly independent eigenvectors. Since the layers grow alternately by and the total number of nodes is: .

Again, since we are growing the tree two layers at a time, we define define

to count the multiplicities of the roots of even indexed polynomial. Similarly, the odd indexed polynomials occur

times at first appearance and also additional times in each larger tree. Thus, we also define .

Counting the multiplicities with the roots of ,the , and gives the following telescoping sum:

which completes the theorem.

4.2. Longer Periods

The analysis for 2–periodic trees can be extended to -periodic trees of any period. The matrices in terms of the shift operator are almost circulant matrices with the first rows taking the form:

and final two rows:

and

In these cases, the recurrence relation of the polynomials associated to the are given by:

where is the sum over all –term products of the with no consecutive indices . For example,

and

With these formulas, constructing the polynomials and counting the eigenvalues proceeds as in the proof of Theorem 8.

4.3. Recurrence Branching

Another interesting class of trees are those where the number of branches at each layer grows with . These admit a particularly simple spectral structure in the limit, described in the following theorem. For example, we could consider the “Fibonacci” tree which has two branches from the root, followed by three, then five, continuing. In this case, due to the exponentially increasing proportion of leaves, the eigenvalue comes to dominate the distribution as recorded in the following result.

Theorem 9.

Let be an increasing sequence. Then, as the proportion of eigenvalues of that are equal to zero goes to 1.

Proof.

Setting , the number of nodes (and equivalently the dimension of the full eigenspace for is equal to . To each set of leaves attached to the same vertex there is a corresponding dimensional zero–eigenspace giving a lower bound on the total dimension of the zero–eigenspace of . Thus, it suffices to show that: