Eigenvalue distribution in random trees Stephan Wagner (joint work - - PowerPoint PPT Presentation
Eigenvalue distribution in random trees Stephan Wagner (joint work with Kenneth Dadedzi) Naples, 13 September 2018 Tree eigenvalues To every graph, and in particular every tree, we can associate an adjacency matrix, a Laplacian matrix, and
Stephan Wagner (joint work with Kenneth Dadedzi) Naples, 13 September 2018
To every graph, and in particular every tree, we can associate an adjacency matrix, a Laplacian matrix, and several other interesting matrices. v1 v2 v3 v4 v5 A = 1 1 1 1 1 1 1 1 L = 1 −1 −1 2 −1 −1 3 −1 −1 −1 1 −1 1
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We will be interested in the eigenvalues of these matrices. v1 v2 v3 v4 v5 A = 1 1 1 1 1 1 1 1 L = 1 −1 −1 2 −1 −1 3 −1 −1 −1 1 −1 1 Eigenvalues of A: −1.84776, −0.765367, 0, 0.765367, 1.84776 Eigenvalues of L: 0, 0.518806, 1, 2.31111, 4.17009
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A tree with 100 vertices (left) and the distribution of the eigenvalues of its adjacency matrix (right).
1 2
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A tree with 100 vertices (left) and the distribution of the eigenvalues of its Laplacian matrix (right).
1 2 3 4 5 6
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Definition
Let T be a fixed tree with n vertices. Picking one of the n eigenvalues (counted with multiplicity) uniformly at random, we obtain a random variable XT : P(XT = x) = multiplicity of x as an eigenvalue of T n . The distribution of this random variable is the spectral distribution of T.
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Definition
Let T be a fixed tree with n vertices. Picking one of the n eigenvalues (counted with multiplicity) uniformly at random, we obtain a random variable XT : P(XT = x) = multiplicity of x as an eigenvalue of T n . The distribution of this random variable is the spectral distribution of T. What can we say about the spectral distribution of a large random tree T?
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The simplest type of model uses the uniform distribution on the set of trees of a given order within a specified family (e.g. the family of all labelled trees, all unlabelled trees or all binary trees).
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The simplest type of model uses the uniform distribution on the set of trees of a given order within a specified family (e.g. the family of all labelled trees, all unlabelled trees or all binary trees). The analysis of such models often involves exact counting and generating functions.
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The simplest type of model uses the uniform distribution on the set of trees of a given order within a specified family (e.g. the family of all labelled trees, all unlabelled trees or all binary trees). The analysis of such models often involves exact counting and generating functions. In particular, this is the case for simply generated families of trees.
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On the set of all rooted ordered (plane) trees, we impose a weight function by first specifying a sequence 1 = w0, w1, w2, . . . and then setting w(T) =
wKi(T)
i
, where Ki(T) is the number of vertices of outdegree i in T. Then we pick a tree of given order n at random, with probabilities proportional to the
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On the set of all rooted ordered (plane) trees, we impose a weight function by first specifying a sequence 1 = w0, w1, w2, . . . and then setting w(T) =
wKi(T)
i
, where Ki(T) is the number of vertices of outdegree i in T. Then we pick a tree of given order n at random, with probabilities proportional to the
w0 = w1 = w2 = · · · = 1 generates random plane trees,
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On the set of all rooted ordered (plane) trees, we impose a weight function by first specifying a sequence 1 = w0, w1, w2, . . . and then setting w(T) =
wKi(T)
i
, where Ki(T) is the number of vertices of outdegree i in T. Then we pick a tree of given order n at random, with probabilities proportional to the
w0 = w1 = w2 = · · · = 1 generates random plane trees, w0 = w2 = 1 (and wi = 0 otherwise) generates random binary trees,
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On the set of all rooted ordered (plane) trees, we impose a weight function by first specifying a sequence 1 = w0, w1, w2, . . . and then setting w(T) =
wKi(T)
i
, where Ki(T) is the number of vertices of outdegree i in T. Then we pick a tree of given order n at random, with probabilities proportional to the
w0 = w1 = w2 = · · · = 1 generates random plane trees, w0 = w2 = 1 (and wi = 0 otherwise) generates random binary trees, wi = 1
i! generates random rooted labelled trees.
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A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set {0, 1, 2, . . .}.
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A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set {0, 1, 2, . . .}. Start with a single vertex, the root.
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A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set {0, 1, 2, . . .}. Start with a single vertex, the root. At time t, all vertices at level t (i.e., distance t from the root) produce a number of children, independently at random according to the fixed distribution (some of the vertices might therefore not have children at all).
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A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set {0, 1, 2, . . .}. Start with a single vertex, the root. At time t, all vertices at level t (i.e., distance t from the root) produce a number of children, independently at random according to the fixed distribution (some of the vertices might therefore not have children at all). A random Galton-Watson tree of order n is obtained by conditioning the process.
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A classical branching model to generate random trees is the Galton-Watson tree model: fix a probability distribution on the set {0, 1, 2, . . .}. Start with a single vertex, the root. At time t, all vertices at level t (i.e., distance t from the root) produce a number of children, independently at random according to the fixed distribution (some of the vertices might therefore not have children at all). A random Galton-Watson tree of order n is obtained by conditioning the process. Simply generated trees and Galton-Watson trees are essentially equivalent. For example, a geometric distribution for branching will result in a random plane tree, a Poisson distribution in a random rooted labelled tree.
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An example: Consider the Galton-Watson process based on a geometric distribution with P(X = k) = pqk (p = 1 − q).
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An example: Consider the Galton-Watson process based on a geometric distribution with P(X = k) = pqk (p = 1 − q). The tree above has probability p7(pq)2(pq2)2(pq3)2 = p13q12,
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An example: Consider the Galton-Watson process based on a geometric distribution with P(X = k) = pqk (p = 1 − q). The tree above has probability p7(pq)2(pq2)2(pq3)2 = p13q12, as does every tree with 13 vertices.
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Another random model that produces very different shapes uses the following simple process, which generates random recursive trees:
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Another random model that produces very different shapes uses the following simple process, which generates random recursive trees: Start with the root, which is labelled 1.
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Another random model that produces very different shapes uses the following simple process, which generates random recursive trees: Start with the root, which is labelled 1. The n-th vertex is attached to one of the previous vertices, uniformly at random.
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Another random model that produces very different shapes uses the following simple process, which generates random recursive trees: Start with the root, which is labelled 1. The n-th vertex is attached to one of the previous vertices, uniformly at random. In this way, the labels along any path that starts at the root are increasing. Clearly, there are (n − 1)! possible recursive trees of order n, and there are indeed interesting connections to permutations. The model can be modified by not choosing a parent uniformly at random, but depending on the current outdegrees (to generate, for example, binary increasing trees).
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An example of an increasing tree: 7 10 13 8 11 12 3 9 6 2 4 5 1
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Theorem (Bhamidi/Evans/Sen, 2012)
Under fairly general assumptions on the random tree model (for example, uniformly random labelled trees and random recursive trees are covered), there exists a (model dependent) deterministic probability distribution S such that the spectral distribution of random n-vertex trees converges in distribution to S in the topology of weak convergence of probability measures on R.
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Theorem (Bhamidi/Evans/Sen, 2012)
Under fairly general assumptions on the random tree model (for example, uniformly random labelled trees and random recursive trees are covered), there exists a (model dependent) deterministic probability distribution S such that the spectral distribution of random n-vertex trees converges in distribution to S in the topology of weak convergence of probability measures on R. Not much is known about the limiting distribution S, not even in special
discrete or contains a continuous component.
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In the following, we will focus on simply generated trees, or equivalently Galton-Watson trees. By Tn, we denote a random tree with n vertices from some given simply generated family (e.g. rooted labelled trees, binary trees).
Theorem
Let α be a fixed real number. The measure of {α} in the spectral distribution of Tn (i.e., the proportion of α among the eigenvalues of Tn) converges in probability to a constant cα ≥ 0 (that depends not only on α, but also the family of trees). If α is an eigenvalue of some tree in the respective family, then cα > 0.
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Theorem
Let α be a fixed real number. The measure of {α} in the spectral distribution of Tn (i.e., the proportion of α among the eigenvalues of Tn) converges in probability to a constant cα ≥ 0 (that depends not only on α, but also the family of trees). If α is an eigenvalue of some tree in the respective family, then cα > 0.
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Theorem
Let α be a fixed real number. The measure of {α} in the spectral distribution of Tn (i.e., the proportion of α among the eigenvalues of Tn) converges in probability to a constant cα ≥ 0 (that depends not only on α, but also the family of trees). If α is an eigenvalue of some tree in the respective family, then cα > 0. An analogous statement holds for Laplacian eigenvalues; in this case, cα > 0 provided that α is an eigenvalue of an augmented Laplacian matrix
entry corresponding to the root in the Laplacian matrix).
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Definition
An invariant F(T) defined for rooted trees T is called additive if it satisfies a recursion of the form F(T) =
k
F(Ti) + f(T), where T1, . . . , Tk are the branches of the tree and f(T) is a so-called toll function.
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Definition
An invariant F(T) defined for rooted trees T is called additive if it satisfies a recursion of the form F(T) =
k
F(Ti) + f(T), where T1, . . . , Tk are the branches of the tree and f(T) is a so-called toll function.
Definition
A fringe subtree of a rooted tree is a subtree consisting of a vertex v and all its descendants. The vertex v is the natural root of this subtree.
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For a vertex v of a rooted tree T, let Tv denote the fringe subtree rooted at v. If F is additive with toll function f, then we have F(T) =
f(Tv). This is easily proved by induction.
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For a vertex v of a rooted tree T, let Tv denote the fringe subtree rooted at v. If F is additive with toll function f, then we have F(T) =
f(Tv). This is easily proved by induction. Thus, if we set f(T) =
T is isomorphic to S,
then the corresponding additive functional F counts the number of
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For us, the relevant functional is Nα(T), the multiplicity of α as an eigenvalue of T. Letting T1, . . . , Tk be the branches of T, the sum
k
Nα(Ti) is the multiplicity of α as an eigenvalue of the disjoint union k
i=1 Ti.
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For us, the relevant functional is Nα(T), the multiplicity of α as an eigenvalue of T. Letting T1, . . . , Tk be the branches of T, the sum
k
Nα(Ti) is the multiplicity of α as an eigenvalue of the disjoint union k
i=1 Ti.
This is also the subgraph of T obtained by removing the root. So Cauchy’s interlacing theorem immediately shows that nα(T) = Nα(T) −
k
Nα(Ti) ∈ {−1, 0, 1}. In other words, the toll function only takes the values 0 and ±1.
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Convergence of the proportion Nα(Tn)/n is now a consequence of the following general theorem due to Janson:
Theorem (Janson 2012)
If the toll function f is bounded, then the associated additive functional F satisfies F(Tn) n
p
→ E(T ), where T denotes an unconditioned Galton-Watson tree.
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Convergence of the proportion Nα(Tn)/n is now a consequence of the following general theorem due to Janson:
Theorem (Janson 2012)
If the toll function f is bounded, then the associated additive functional F satisfies F(Tn) n
p
→ E(T ), where T denotes an unconditioned Galton-Watson tree. The constant E(T ) is clearly positive in the special case “occurrences of a fixed fringe subtree”. This will be exploited in the following construction.
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Let α be an eigenvalue of some rooted tree R, and consider a tree S consisting of a root to which two (or possibly more) copies of R are attached.
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Let α be an eigenvalue of some rooted tree R, and consider a tree S consisting of a root to which two (or possibly more) copies of R are attached. Now suppose S occurs k times as a fringe subtree in a larger tree T.
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Let α be an eigenvalue of some rooted tree R, and consider a tree S consisting of a root to which two (or possibly more) copies of R are attached. Now suppose S occurs k times as a fringe subtree in a larger tree T. When the roots of these k fringe subtrees are removed, at least 2k of the remaining components are isomorphic to R. Thus the multiplicity
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Let α be an eigenvalue of some rooted tree R, and consider a tree S consisting of a root to which two (or possibly more) copies of R are attached. Now suppose S occurs k times as a fringe subtree in a larger tree T. When the roots of these k fringe subtrees are removed, at least 2k of the remaining components are isomorphic to R. Thus the multiplicity
By the interlacing property, the multiplicity of α as an eigenvalue of T is at least k.
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Let α be an eigenvalue of some rooted tree R, and consider a tree S consisting of a root to which two (or possibly more) copies of R are attached. Now suppose S occurs k times as a fringe subtree in a larger tree T. When the roots of these k fringe subtrees are removed, at least 2k of the remaining components are isomorphic to R. Thus the multiplicity
By the interlacing property, the multiplicity of α as an eigenvalue of T is at least k. Thus the multiplicity of α is always greater than or equal to the number of
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By the aforementioned result of Janson, we have cα = E(nα(T )) = P(nα(T ) = 1) − P(nα(T ) = 1). An analytic expression can be obtained by means of generating functions.
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Recall that a family of simply generated trees is characterised by a weight function on rooted ordered trees: w(T) =
wKi(T)
i
, for a given weight sequence 1 = w0, w1, w2, . . ..
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Recall that a family of simply generated trees is characterised by a weight function on rooted ordered trees: w(T) =
wKi(T)
i
, for a given weight sequence 1 = w0, w1, w2, . . .. This can be translated to an identity for the (weighted) generating function T(x) =
w(T)x|T|, the sum being over all rooted ordered trees T. The coefficient of xn is the total weight of all trees with n vertices.
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We have T(x) =
w(T)x|T|
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We have T(x) =
w(T)x|T| =
· · ·
wkw(T1) · · · w(Tk)x|T1|+···+|Tk|+1
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We have T(x) =
w(T)x|T| =
· · ·
wkw(T1) · · · w(Tk)x|T1|+···+|Tk|+1 = x
wk
T1
w(T1)x|T1|k
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We have T(x) =
w(T)x|T| =
· · ·
wkw(T1) · · · w(Tk)x|T1|+···+|Tk|+1 = x
wk
T1
w(T1)x|T1|k = x
wkT(x)k
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We have T(x) =
w(T)x|T| =
· · ·
wkw(T1) · · · w(Tk)x|T1|+···+|Tk|+1 = x
wk
T1
w(T1)x|T1|k = x
wkT(x)k = xΦ(T(x)).
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We have T(x) =
w(T)x|T| =
· · ·
wkw(T1) · · · w(Tk)x|T1|+···+|Tk|+1 = x
wk
T1
w(T1)x|T1|k = x
wkT(x)k = xΦ(T(x)). Here, Φ(t) =
k≥0 wktk is the weight generating function associated with
the specific family. For example, Φ(t) = et for rooted labelled trees, Φ(t) =
1 1−t for rooted ordered (plane) trees, or Φ(t) = 1 + t2 for binary
trees.
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If there exists a positive real solution τ to the equation Φ(t) = tΦ′(t), then
singularity at ρ = τ/Φ(τ) = 1/Φ′(τ), and a lot of information (in particular, an asymptotic formula for the number/total weight of n-vertex trees) can be inferred.
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If there exists a positive real solution τ to the equation Φ(t) = tΦ′(t), then
singularity at ρ = τ/Φ(τ) = 1/Φ′(τ), and a lot of information (in particular, an asymptotic formula for the number/total weight of n-vertex trees) can be inferred. For us, it is mostly important that the constant cα can be expressed as cα = 1 τ
w(T)nα(T)ρ|T| = 1 τ
w(T)ρ|T| −
w(T)ρ|T| .
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In order to evaluate the expression, we will categorise rooted trees. For a rooted tree T with root r, let Ψ(T, z) = det(zI − A(T)) be the characteristic polynomial of (the adjacency matrix of) T, and let Ψr(T, z) be the characteristic polynomial of (the adjacency matrix of) the forest that results when the root is removed.
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In order to evaluate the expression, we will categorise rooted trees. For a rooted tree T with root r, let Ψ(T, z) = det(zI − A(T)) be the characteristic polynomial of (the adjacency matrix of) T, and let Ψr(T, z) be the characteristic polynomial of (the adjacency matrix of) the forest that results when the root is removed. Consider the quotient Q(T, z) = Ψr(T, z) Ψ(T, z) .
Eigenvalue distribution in random trees
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In order to evaluate the expression, we will categorise rooted trees. For a rooted tree T with root r, let Ψ(T, z) = det(zI − A(T)) be the characteristic polynomial of (the adjacency matrix of) T, and let Ψr(T, z) be the characteristic polynomial of (the adjacency matrix of) the forest that results when the root is removed. Consider the quotient Q(T, z) = Ψr(T, z) Ψ(T, z) . We note that nα(T) = 1 if and only if Q(T, z) has a pole at z = α,
Eigenvalue distribution in random trees
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In order to evaluate the expression, we will categorise rooted trees. For a rooted tree T with root r, let Ψ(T, z) = det(zI − A(T)) be the characteristic polynomial of (the adjacency matrix of) T, and let Ψr(T, z) be the characteristic polynomial of (the adjacency matrix of) the forest that results when the root is removed. Consider the quotient Q(T, z) = Ψr(T, z) Ψ(T, z) . We note that nα(T) = 1 if and only if Q(T, z) has a pole at z = α, nα(T) = −1 if and only if Q(T, z) has a zero at z = α,
Eigenvalue distribution in random trees
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In order to evaluate the expression, we will categorise rooted trees. For a rooted tree T with root r, let Ψ(T, z) = det(zI − A(T)) be the characteristic polynomial of (the adjacency matrix of) T, and let Ψr(T, z) be the characteristic polynomial of (the adjacency matrix of) the forest that results when the root is removed. Consider the quotient Q(T, z) = Ψr(T, z) Ψ(T, z) . We note that nα(T) = 1 if and only if Q(T, z) has a pole at z = α, nα(T) = −1 if and only if Q(T, z) has a zero at z = α, nα(T) = 0 if and only if limz→α Q(T, z) is finite and not zero.
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Let T be a rooted tree with root r, and let T1, . . . , Tk be its branches with roots r1, . . . , rk. It is not too hard to show that Ψr(T, z) Ψ(T, z) = 1 z − k
j=1 Ψrj (Tj,z) Ψr(Tj,z)
,
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Let T be a rooted tree with root r, and let T1, . . . , Tk be its branches with roots r1, . . . , rk. It is not too hard to show that Ψr(T, z) Ψ(T, z) = 1 z − k
j=1 Ψrj (Tj,z) Ψr(Tj,z)
,
Q(T, z) = 1 z − k
j=1 Q(Tj, z)
.
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Let T be a rooted tree with root r, and let T1, . . . , Tk be its branches with roots r1, . . . , rk. It is not too hard to show that Ψr(T, z) Ψ(T, z) = 1 z − k
j=1 Ψrj (Tj,z) Ψr(Tj,z)
,
Q(T, z) = 1 z − k
j=1 Q(Tj, z)
. We find that Q(T, z) has a zero at z = α if and only if at least one of the Q(Tj, z) has a pole (one direction is clear, for the converse one needs to ensure that there cannot be cancellations in the denominator), Q(T, z) has a pole at z = α if and only if k
j=1 Q(Tj, α) = α,
Q(T, α) = β if and only if k
j=1 Q(Tj, α) = α − 1/β.
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Thus we have a recursive description of trees according to their “type” (zero, pole or other value). For β ∈ R ∪ {∞}, we set Hβ(x) =
w(T)x|T|.
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Thus we have a recursive description of trees according to their “type” (zero, pole or other value). For β ∈ R ∪ {∞}, we set Hβ(x) =
w(T)x|T|. Clearly, we have T(x) =
Hβ(x).
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Thus we have a recursive description of trees according to their “type” (zero, pole or other value). For β ∈ R ∪ {∞}, we set Hβ(x) =
w(T)x|T|. Clearly, we have T(x) =
Hβ(x). Moreover, cα = 1 τ
w(T)ρ|T| −
w(T)ρ|T| = 1 τ (H∞(ρ) − H0(ρ)).
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The recursive characterisation of Q(T, z) can be turned into the following system of functional equations: H0(x) = xΦ(T(x)) − xΦ(T(x) − H∞(x)), Hβ(x) = [yα−1/β]Φ
γ∈R
Hγ(x)yγ . In the second equation, y is just a formal variable, and there are actually infinitely many equations (one for each β).
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The infinite system of equations can actually be solved numerically, and yields results such as the following:
Theorem
The value of the constant c1 (limiting proportion of the eigenvalue 1) is ≈ 0.0213 for random (rooted) labelled trees, ≈ 0.0255 for random plane trees, ≈ 0.0141 for random pruned binary trees.
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The infinite system of equations can actually be solved numerically, and yields results such as the following:
Theorem
The value of the constant c1 (limiting proportion of the eigenvalue 1) is ≈ 0.0213 for random (rooted) labelled trees, ≈ 0.0255 for random plane trees, ≈ 0.0141 for random pruned binary trees. In other words, the eigenvalue 1 makes up about 2.13% (2.55%, 1.41%) of the spectrum of large random labelled (plane, pruned binary) trees, respectively.
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Making use of ideas of Drmota, Gittenberger and Morgenbesser on infinite systems of functional equations, one also obtains a central limit theorem, i.e. a result of the following form:
Theorem
For a given simply generated family of trees and a real number α that
cα > 0 and bα > 0 such that mean and variance of Nα(Tn) are equal to µn = cαn + o(n) and σ2
n = bαn + o(n) respectively, and the normalised
random variable Nα(Tn) − µn
n
converges weakly to a standard normal distribution.
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The eigenvalue 0 is special in some sense: n0(T) = ±1 for all trees T (there are no trees with n0(T) = 0), so there are only two possibilities: Q(T, z) has either a pole or a zero at z = 0.
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The eigenvalue 0 is special in some sense: n0(T) = ±1 for all trees T (there are no trees with n0(T) = 0), so there are only two possibilities: Q(T, z) has either a pole or a zero at z = 0. There is also a combinatorial interpretation: since the characteristic polynomial of the adjacency matrix coincides with the matching polynomial for all forests, N0(T) = n − 2µ(T), where n is the number of vertices and µ(T) the matching number.
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The system of functional equations reduces to a 2 × 2-system, and explicit values of c0 can be found for some families of trees: c0 = 2W(1) − 1 ≈ 0.134 for random (rooted) labelled trees (where W is the Lambert W-function), c0 = √ 5 − 2 ≈ 0.236 for random plane trees, c0 = 7 − 4 √ 3 ≈ 0.0717 for random pruned binary trees.
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Similar results can also be obtained for other classes of trees and other types of spectra, for instance: a central limit theorem for the multiplicity of Laplacian eigenvalues in simply generated trees (based on very similar functional equations); a central limit theorem for the multiplicity Nα in recursive trees; the constant c0 in this case is 1 1 + log t 1 − log t dt ≈ 0.193; a “forcing subtrees” type result for the distance spectrum, showing that the proportion of certain eigenvalues in the distance spectrum of large random rooted trees can be bounded below by positive constants.
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