GGROWTH OF BILINEAR MAPS
VUONG BUI
Abstract.
For a bilinear map ∗ : R d × R d → R d of nonnegative coefficients and a vector s ∈ R d of positive entries, among an exponentially number of ways combining n instancesof s using n − ∗ for a given n , we are interested in the largest entry overall the resulting vectors. An asymptotic behavior is that the n -th root of this largest entryconverges to a growth rate λ when n tends to infinity. In this paper, we prove the existenceof this limit by a special structure called linear pattern. We also pose a question on thepossibility of a relation between the structure and whether λ is algebraic. Introduction
Given a binary operation ∗ and a fixed operand s , we have a variety of ways to combine n instances of s using n − ∗ . The results may vary as the the operation ∗ isnot necessarily commutative or associative. However, we might still expect that the “largestvalue” of all the combinations does not grow too arbitrarily. A problem of this type wasposed in [1] by G¨unter Rote, where ∗ is a bilinear map of nonnegative coefficients and s is avector of positive entries, both in the same vector space. In this paper, the largest entry ofa resulting vector will be shown to be of exponential order with a fixed growth rate.Consider a vector s ∈ R d of all positive entries s i and a bilinear map ∗ : R d × R d → R d reprensented by nonnegative coefficients c k ( i,j ) in the way: If v = u ∗ w then v k = (cid:80) i,j c ( k ) i,j u i w j .Let A n for an integer n ≥ n − ∗ to n instances of s , that is A = { s } and A n = (cid:91) ≤ m ≤ n − { x ∗ y : x, y ∈ A n × A n − m } . Let g ( n ) denote the largest entry over all vectors in A n , that is g ( n ) = max { x i : x ∈ A n , ≤ i ≤ d } . For later convenient usage we also denote by g i ( n ) the largest i -th entry over all vectorsin A n , that is g i ( n ) = max { x i : x ∈ A n } . The growth rate λ of the system is defined as λ = lim n →∞ n (cid:112) g ( n ) . We will prove the validity of this limit and give further discussion after introducing somedefinitions related to a special structure called linear pattern . The author is supported by the Deutsche Forschungsgemeinschaft (DFG) Graduiertenkolleg “Facets ofComplexity” (GRK 2434). a r X i v : . [ c s . D M ] M a y he computation of A n can be related to the binary trees of n leaves. For each binarytree of n leaves, we assign an element in A n as follows: If the tree is only a single leaf thenthe result is s ; otherwise, the result is x ∗ y where x, y are the results corresponding to theleft and right branches of the root, respectively. Every element of v ∈ A n can be computedfrom such a binary tree. Given a tree T , the corresponding vector in A n is said to be thevector obtained by T . Although in principle there may be more than one trees resulting inthe same v , we assign some arbitrary tree for each v . Later arguments are independent ofthe choice.We call a pair of a tree T and a marked leaf (cid:96) of T a linear pattern P = ( T, (cid:96) ). Thisdefinition has some interesting properties.
Proposition 1.
Given a linear pattern P = ( T, (cid:96) ) , if one replaces s by u specifically onlyfor the leaf (cid:96) , then the vector v obtained by T is related to u by a matrix M = M ( P ) suchthat v = M u.
This fact follows from a property of bilinear maps: If we fix one of the two terms of theinput, the new map will be linear, that is: ∗ y ( x ) = x ∗ y and ∗ x ( y ) = x ∗ y are both linear. (cid:96)T = TT T A sequence of trees { T t } t ≥ is said to be generated by a pattern ( T, (cid:96) ) if T = T and T t for t ≥ T by replacing (cid:96) by T t − (seeFigure 1 for example). Proposition 2.
For a linear pattern P = ( T, (cid:96) ) , let h ( t ) be the largest entryof the vector obtained from the tree T t , then the so-called rate of pattern Pλ P = lim t →∞ t (cid:112) h ( t ) is valid and equals to the dominant eigenvalue of the matrix M ( P ) . This is a well known fact and that eigenvalue is often called PerronFrobe-nius eigenvalue or spectral radius.The tree T t has t ( | T | −
1) + 1 leaves where | T | is the number of leaves of T . While h ( t ) is a lower bound for a subsequence of g ( n ), the correspondinglower bound for the rate should be the ( | T | − λ P of λ P instead. Proposition 3.
For every linear pattern P = ( T, (cid:96) ) , lim inf n →∞ n (cid:112) g ( n ) ≥ ¯ λ P . Indeed, let m = | T | −
1, for each n = mp + q (0 ≤ q < m ), consider the tree obtained from T p by replacing the marked leaf by any tree of q leaves (whose evaluation can be seen to bebounded). It is not hard to see that the n -th root of the largest entry obtained from thesetrees converges to ¯ λ P .Moreover, we give the following stronger conclusion, which serves as the proof of thevalidity of λ . Note that we sacrifice precision for a compact notation. In fact, M also depends on ∗ , s , which are fixedfrom the beginning while only pattern P is varied. Therefore, M ( P ) can be clearly understood from contextas M ( P, ∗ , s ) λ P is a shorten form of a more precise notation λ P, ∗ ,s , see footnote 1. heorem 1. The n -th root of g ( n ) converges when n tends to infinity and the limit is thesupremum of ¯ λ P over all patterns P , that is λ = lim n →∞ n (cid:112) g ( n ) = sup P ¯ λ P . There are some cases that λ > ¯ λ P for all P . The following system is one example. Theorem 2. If s = (1 , and x ∗ y = ( x y + x y , x y ) , then λ > ¯ λ P for every P . The optimal trees for this system are perfect binary trees (for n being powers of 2), whichare not recognized by any linear pattern. Actually the system in the above theorem wasstudied in a different formulation (see [2]) and the growth rate was shown to be λ = exp( (cid:88) i ≥ i log(1 + 1 x i )) = 1 . . . . where x n is a sequence with x = 1 and x k +1 = 1 + x k for k ≥ x n (and g (2 n )) is the number of binary trees of heights at most n + 1 (Sequence A λ in this system seems to be not an algebraic number. Since the growthrate is algebraic whenever a pattern recognizes it (and the coefficients and the entries areintegers), Theorem 2 suggests the question of the other direction: Question 1.
Suppose the coefficients of ∗ and the entries of v are all integers. Is it truethat: If λ is algebraic, then there exists a pattern P such that ¯ λ P = λ ? It makes sense to give an example where a pattern recognizes the rate, and hence, thegrowth rate is algebraic.
Theorem 3. If s = (1 , and x ∗ y = ( x y + x y , x y ) , then the growth rate λ is the golden ratio φ , which is recognized by a pattern. In particular, g ( n ) = F n +1 and g ( n ) = F n , where F is the Fibonacci sequence starting with F = F = 1 . The optimal trees are binary trees where every left (or right) branch of every non-leafvertex is just a leaf. The proof uses some inequalities involving the elements of the Fibonaccisequence, which are interesting on its own.The readers may notice that although the two examples in Theorem 2 and Theorem 3just slightly differ from each other, the growth rates and the patterns are of quite differentnatures.A more sophisicated example can be found in [1] with growth rate √
95 and a complexlinear pattern. It actually solves a problem on the maximal number of minimal dominatingsets in a tree. One can also find a proof of the validity of λ for that particular case there (byshowing that g ( n ) is supermultiplicative and then applying Fekete’s lemma [4]).We give the proofs of Theorems 1, 2, 3 in Sections 2, 4, 5, respectively. Section 3 provessome lemmas used in Section 2. . Proof of Theorem 1
Consider the dependency graph that is a directed graph whose vertices are { , . . . , d } ;there is a directed edge from k to i if and only if c ( k ) i,j or c ( k ) j,i is positive, where c ( k ) i,j are thecoefficients of ∗ . We say k depends on i for such an edge ki . In some cases we need to sayspecifically that k left depends (resp. right depends ) on i if c ( k ) i,j (resp. c ( k ) j,i ) is positive.The dependency graph can be partitioned into strongly connected components, which canbe partially ordered. Component C is said to be greater than Component C if eitherthere is a directed edge ij for i ∈ C , j ∈ C , or there exists another component C so that C > C > C .We give some useful lemmas, which will be proved later in Section 3. Lemma 1.
For every i , g i ( n ) is at least a constant time of g i ( n + 1) . Lemma 2. If i, j are of the same component, then lim inf n →∞ n (cid:112) g i ( n ) = lim inf n →∞ n (cid:113) g j ( n ) , lim sup n →∞ n (cid:112) g i ( n ) = lim sup n →∞ n (cid:113) g j ( n ) . If i ∈ C , j ∈ C and C < C then lim inf n →∞ n (cid:112) g i ( n ) ≤ lim inf n →∞ n (cid:113) g j ( n ) , lim sup n →∞ n (cid:112) g i ( n ) ≤ lim sup n →∞ n (cid:113) g j ( n ) . Lemma 3.
Given a pattern P = ( T, (cid:96) ) with its matrix M . Let i, j be two vertices of thesame component, then there exists a pattern P (cid:48) = ( T (cid:48) , (cid:96) (cid:48) ) with the difference in the numberof leaves | T (cid:48) | − | T | bounded and λ P (cid:48) at least a constant time of M i,j . Lemma 4. If M = M ( P ) is the matrix for a pattern P = ( T, (cid:96) ) with T having n leaves,then for every i, j , the value M i,j is at most a constant time of g i ( n ) . Lemma 5.
If a component C is greater than every other component, then g i ( n ) is at least aconstant time of g ( n ) for every i ∈ C . Lemma 6.
For a tree of n > leaves, there is a subtree of m leaves such that n/ ≤ m ≤ n/ . We are now ready to prove Theorem 1.Take any component C , we investigate the C -subsystem, which is the system after exclud-ing all but the dimensions in the components smaller than or equal to C . This restrictionactually does not lose the generality but gives a conclusion on the convergence of n (cid:112) g i ( n )for every i , as we will show later.In the C -subsystem, let λ CP and ¯ λ CP denote the rates with respect to the C -subsystem.It can be seen that lim inf n →∞ n (cid:112) g i ( n ) ≥ sup P ¯ λ CP for i ∈ C by Proposition 3 and Lemma5. We prove the other direction: lim sup n →∞ n (cid:112) g i ( n ) ≤ sup P ¯ λ CP . uppose C is a component C such that(1) lim sup n →∞ n (cid:112) g i ( n ) > lim sup n →∞ (cid:113) g j ( m )for every C (cid:48) < C , i ∈ C , j ∈ C (cid:48) .Let i be a vertex in C and denote θ = lim sup n →∞ n (cid:112) g i ( n ).Then for every (cid:15) >
0, there exist an n (cid:15) such that for every n > n (cid:15) , g i ( n ) < ( θ + (cid:15) ) n , andfor every N there exists n > N such that g i ( n ) > ( θ − (cid:15) ) n .Let k be a vertex and denote θ (cid:48) = lim sup n →∞ n (cid:112) g k ( n ).Then for every (cid:15) >
0, there exists an n (cid:15) such that for every m > n (cid:15) , g k ( m ) < ( θ (cid:48) + (cid:15) ) m .Fix (cid:15) , choose n (cid:15) that works for i and every k , that is for all n > n (cid:15) we have g i ( n ) < ( θ + (cid:15) ) n and g k ( n ) < ( θ (cid:48) + (cid:15) ) n . Let N = 3 n (cid:15) and take any n > N such that g i ( n ) > ( θ − (cid:15) ) n .Consider the tree T corresponding to g i ( n ). Take a subtree T of m leaves with n/ ≤ m ≤ n/ (cid:96) among these m leaves with T to obtaina pattern P = ( T , (cid:96) ). Denote by (cid:96) the root of T , and by T the tree obtained from T after contracting T to (cid:96) . We have another pattern P = ( T , (cid:96) ). Also, consider the pattern P = ( T, (cid:96) ) for (cid:96) = (cid:96) .Let the matrices for P, P , P be M, A, B , respectively. Clearly, M = AB.
Since g i ( n ) = (cid:80) j M i,j s j , there exists some j such that M i,j ≥ const g i ( n ) . Since M i,j = (cid:80) k A i,k B k,j , there exists k such that A i,k B k,j ≥ const M i,j ≥ const g i ( n ) ≥ const( θ − (cid:15) ) n . By Lemma 4, and by the definition of θ (cid:48) with m > n (cid:15) , B k,j ≤ const g k ( m ) ≤ const( θ (cid:48) + (cid:15) ) m . It means A i,k is at least a constant time of( θ − (cid:15) ) n ( θ (cid:48) + (cid:15) ) m = ( θ − (cid:15)θ (cid:48) + (cid:15) ) m ( θ − (cid:15) ) n − m ≥ ( (cid:114) θ − (cid:15)θ (cid:48) + (cid:15) ) n − m ( θ − (cid:15) ) n − m = ( (cid:114) θ − (cid:15)θ (cid:48) + (cid:15) θ − (cid:15)θ + (cid:15) ) n − m ( θ + (cid:15) ) n − m . The inequality step is due to m ≥ ( n − m ) / k is of a smaller component than C , that is θ (cid:48) < θ .When (cid:15) is small and n is large enough, the value of A i,k will be not bounded by a constanttime of ( θ + (cid:15) ) n − m due to ( θ − (cid:15) ) / ( θ (cid:48) + (cid:15) ) > θ − (cid:15) ) / ( θ + (cid:15) ) tending to 1 when (cid:15) tendsto 0. However, A i,k ≤ const g i ( n − m + 1) ≤ const g i ( n − m ) ≤ const( θ + (cid:15) ) n − m by Lemma 4and Lemma 1, a contradiction (note that T has n − m + 1 leaves). herefore, i and k are of the same component, which means B k,j is at most a constanttime of ( θ + (cid:15) ) m . It follows that A i,k is at least a constant time of( θ − (cid:15) ) n ( θ + (cid:15) ) m . For every (cid:15) (cid:48) > (cid:15) > θ − (cid:15) ) n ( θ + (cid:15) ) m > ( θ − (cid:15) (cid:48) ) n − m . By Lemma 3, the lower bound of A i,k shows that there exists a pattern P (cid:48) having ¯ λ C P (cid:48) greater than a number arbitrarily close to θ from below when (cid:15) (cid:48) gets smaller. In other words,lim sup n →∞ n (cid:112) g i ( n ) ≤ sup P ¯ λ C P . It means lim n →∞ n (cid:112) g i ( n ) exists for every i ∈ C since the limit superior and the limitinferior are equal.We have shown that n (cid:112) g i ( n ) converges to a limit for every i in a component satisfyingthe requirement (1). It remains to consider components C not satisfying the requirement.For such a component C , there is a component C < C satisfying that requirement andlim sup n →∞ n (cid:112) g i ( n ) = lim sup n →∞ n (cid:112) g k ( n ) for any i ∈ C and k ∈ C . By Lemma 2,lim inf n →∞ n (cid:112) g k ( n ) ≥ lim inf n →∞ n (cid:112) g i ( n ) = lim sup n →∞ n (cid:112) g i ( n ) = lim sup n →∞ n (cid:112) g k ( n ) . It means lim n →∞ n (cid:112) g k ( n ) exists because the limit superior and limit inferior are equal.The existence of λ = lim n →∞ n (cid:112) g ( n ) = max k lim n →∞ n (cid:112) g k ( n )follows from the existence of lim n →∞ n (cid:112) g k ( n ) for every k .This limit λ equals to the supremum of ¯ λ P over all patterns P because for i ∈ C satisfyinglim n →∞ n (cid:112) g i ( n ) = λ , we havesup P ¯ λ CP ≤ sup P ¯ λ P ≤ lim n →∞ n (cid:112) g ( n ) = lim n →∞ n (cid:112) g i ( n ) = sup P ¯ λ CP . Proofs of the lemmas
Proof of Lemma 1.
Let T be the tree corresponding to g i ( n + 1). Take any subtree T of 2leaves, and replace it by a leaf, denoted by (cid:96) , to obtain a new tree T (cid:48) of n leaves.Let v, v (cid:48) be the vector obtained by the trees T, T (cid:48) , respectively.Let M be the matrix for the pattern ( T (cid:48) , (cid:96) ), that is v (cid:48) = M s for the vector s at the leaf (cid:96) .If the leaf (cid:96) is replaced by the tree T , we have the relation v = M u where u = s ∗ s is thevector obtained by T .Since u i ≤ g (2) and s i ≥ min k s k for every i , u i s i ≤ g (2)min k s k . We can write u ≤ ( g (2) / min k s k ) s . ogether with v = M u and v (cid:48) = M s , we have v i v (cid:48) i ≤ g (2)min k s k . The conclusion follows due to v (cid:48) i ≤ g i ( n ). (cid:3) Remark 1.
It is possible to obtain a more general conclusion by choosing T of more thantwo leaves. However, we cannot guarantee the size of T in this case but only some bound onit (as in Lemma 6). The question is: Is it true that g ( n ) ≤ const g ( p ) g ( q ) for every p, q ≥ , p + q = n ? The validity of λ just follows if this is true (by Fekete’s lemma). Before proving the remaining lemmas, we give the following useful corollary of Lemma 1.
Corollary 1.
Given a fixed d , for every i , g i ( n ) is at least a constant time of g i ( n + d ) .Proof of Lemma 2. Suppose there is an edge ki in the dependency graph. For each n , let T be the tree corresponding to g i ( n ). Consider the tree T (cid:48) of n + 1 leaves where the left (resp.right) branch of the root is T if k left (resp. right) depends on i , and the other branch isjust a single leaf. It can be seen from T (cid:48) that g k ( n + 1) ≥ const g i ( n ) . Suppose i, j be two vertices so that there exists a path of length d from j to i , we have g j ( n + d ) ≥ const g i ( n ) . By Corollary 1, g j ( n + d ) ≤ const g j ( n ), therefore,(2) g j ( n ) ≥ const g i ( n ) . If there is a path from j to i , it follows from Equation (2) thatlim inf n →∞ n (cid:112) g i ( n ) ≤ lim inf n →∞ n (cid:113) g j ( n ) , lim sup n →∞ n (cid:112) g i ( n ) ≤ lim sup n →∞ n (cid:113) g j ( n ) . It is indeed the case when i ∈ C , j ∈ C and C < C .If i, j are of the same component, then there exist a path from i to j and also a path from j to i . Apply the above inequalities to both i, j and j, i , we obtainlim inf n →∞ n (cid:112) g i ( n ) = lim inf n →∞ n (cid:113) g j ( n ) , lim sup n →∞ n (cid:112) g i ( n ) = lim sup n →∞ n (cid:113) g j ( n ) . (cid:3) Proof of Lemma 3.
Let the path from j to i be k , . . . , k d for k = j , k d = i and the lengthof the path d . Construct the trees T , . . . , T d such that T d = T , and for t < d , one of the twobranches of the root of T t is T t +1 and the other is just a single leaf. If k t left (right) dependson k t +1 then the branch of T t +1 is on the left (right) in T t . et P (cid:48) be the pattern ( T (cid:48) , (cid:96) (cid:48) ) for T (cid:48) = T , (cid:96) (cid:48) = (cid:96) , and M (cid:48) the matrix of P (cid:48) . We can seethat | T (cid:48) | − | T | is bounded and M (cid:48) j,j is at least a constant time of M i,j . It follows that λ P (cid:48) isat least a constant time of M i,j since λ P (cid:48) ≥ M (cid:48) j,j . (cid:3) Proof of Lemma 4.
Let the vector obtained from T be v = M s . Since v i = (cid:80) j M i,j s j and v i ≤ g i ( n ), the value M i,j for any j is at most a constant time of g i ( n ). (cid:3) Proof of Lemma 5.
Since there is a path from i to j for every i ∈ C and any other j , by thesame argument as in Equation (2), we have g i ( n ) ≥ const g j ( n ) . If we do not fix j , we still have g i ( n ) ≥ const max j g j ( n ) . The conclusion follows since for each n , g ( n ) = g j ( n ) for some j . (cid:3) Proof of Lemma 6.
Assume there is no such subtree. Consider the two branches of the root,one of them has less than n/ n/ n/ n/ n/ n/ n leaves, a contradiction. (cid:3) Proof of Theorem 2
We know (see [2]) that the growth rate for this system can be attained by perfect binarytrees (no linear pattern recognizes them). We will show further that there is no linear patternof the rate by verifying that given any linear pattern, we can construct another pattern of ahigher rate.Consider a pattern P = ( T, (cid:96) ) with the corresponding matrix (cid:20) a bc d (cid:21) . It is verifiable that a, b are always at least 1 while c is always 0 and d is always 1 (thereaders can check themselves or just see it by the manipulations of patterns and matricesthroughout the proof). The dominant eigenvalue of the matrix is always a since c = 0 and d = 1. The rate is therefore the m -th root of a where m is one less than the number of leavesin T .Suppose we have two patterns of P = ( T , (cid:96) ) and P = ( T , (cid:96) ) with their matricesrespectively (cid:20) a b (cid:21) , (cid:20) a b (cid:21) . Their product is (cid:20) a a a b + b (cid:21) . If we construct a new pattern P = ( T, (cid:96) ) with T obtained from T by replacing (cid:96) by T and let (cid:96) be (cid:96) , then ¯ λ P ≤ max { ¯ λ P , ¯ λ P } , since the dominant eigenvalue of the product is a a . It means that we do not need toconsider patterns that are decomposable into two patterns in the above way for a candidate f the best rate. In other words, the maximal ¯ λ P if exists can be found among the patternswhere one branch of the root is just a leaf, which is also the marked leaf.Let the other branch than the branch of the marked leaf has the evaluation ( a, (cid:20) a
10 1 (cid:21) has the dominant eigenvalue a .So, the growth rate of the system is λ = sup n n (cid:112) g ( n ) . However, we do not have any n so that n (cid:112) g ( n ) = λ . Suppose the contrary, let T be atree of n leaves whose first entry has value g ( n ) attaining the maximum rate λ . Let T (cid:48) be atree where each branch of the root is a copy of T . The first entry of the evaluation of T (cid:48) is( g ( n )) + 1, but T (cid:48) has 2 n leaves, hence T (cid:48) attains a higher rate than λ , a contradiction.So, for every linear pattern, we always obtain another pattern of a higher rate. Theconclusion of Theorem 2 follows.5. Proof of Theorem 3
We will show that the growth rate of this system is the golden ratio by showing that g ( n ) , g ( n ) are respectively F n +1 , F n , where F is the Fibonacci sequence with starting ele-ments F = 0 , F = 1 , F = 1.It can be seen that g ( n ) ≥ F n +1 and g ( n ) ≥ F n for every n ≥ F n +1 , F n )is the evaluation of the tree T n − for the pattern ( T, (cid:96) ) where T is the tree of two leaves, anyof them can be chosen as the marked leaf (cid:96) .In order to show that they are also the upper bounds, we prove following lemma. Lemma 7.
Let F be the Fibonacci sequence with starting elements F = 0 , F = 1 , F = 1 ,then the following inequalities F p F q − + F p − F q ≤ F p + q − ,F p F q ≤ F p + q − hold for every p, q ≥ Proof.
The conclusion holds for any ( p, q ) ∈ ( { , } × N + ) ∪ ( N + × { , } ), i.e. one of thefour conditions p = 1, p = 2, q = 1, q = 2 holds.As for the first inequality, if p = 1 (similarly for q = 1), then the inequality is equivalentto F q − ≤ F q . If p = 2 (similarly for q = 2), then it is equivalent to F q − + F q ≤ F q +1 .As for the second inequality, if p = 1 (similarly for q = 1), then the inequality is equivalentto F q ≤ F q . If p = 2 (similarly for q = 2), then it is equivalent to F q ≤ F q +1 .We prove the lemma by induction. Suppose the inequalities hold for any ( p (cid:48) , q (cid:48) ) ∈ { p − , p − } × { q − , q − } , we show that they also hold for ( p, q ). ndeed, F p F q − + F p − F q = ( F p − + F p − )( F q − + F q − ) + ( F p − + F p − )( F q − + F q − )= ( F p − F q − + F p − F q − ) + ( F p − F q − + F p − F q − )+ ( F p − F q − + F p − F q − ) + ( F p − F q − + F p − F q − ) ≤ F p + q − + F p + q − + F p + q − + F p + q − = F p + q − + F p + q − = F p + q − , and F p F q = ( F p − + F p − )( F q − + F q − )= F p − F q − + F p − F q − + F p − F q − + F p − F q − ≤ F p + q − + F p + q − + F p + q − + F p + q − ≤ F p + q − + F p + q − ≤ F p + q − . By induction, the inequalities hold for every p, q ≥ (cid:3) Now the verification for the upper bounds of g ( n ) and g ( n ) becomes clear. They holdtrivially for n = 1. For higher n , if g ( n ) corresponds to a tree where the left branch of theroot has p leaves and the right branch has q leaves ( p + q = n ), then the same bounds hold: g ( n ) ≤ g ( p ) g ( q ) + g ( p ) g ( q ) = F p +1 F q + F p F q +1 ≤ F p + q +1 = F n +1 , and g ( n ) ≤ g ( p ) g ( q ) = F p +1 F q ≤ F p + q = F n . Being both lower bounds and upper bounds, we have g ( n ) = F n +1 and g ( n ) = F n . Acknowledgement
The author would like to thank G¨unter Rote for introducing the problem and reading theproofs, Roman Karasev for his suggestions to some improvements in the presentation.
References [1] G¨unter Rote. The maximum number of minimal dominating sets in a tree. In
Proceedings of the ThirtiethAnnual ACM-SIAM Symposium on Discrete Algorithms , pages 1201–1214. SIAM, 2019.[2] Anna de Mier and Marc Noy. On the maximum number of cycles in outerplanar and series–parallelgraphs.
Graphs and Combinatorics , 28(2):265–275, 2012.[3] Alfred V Aho and Neil JA Sloane. Some doubly exponential sequences.
Fibonacci Quart , 11(4):429–437,1973.[4] Michael Fekete. ¨Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligenKoeffizienten.
Mathematische Zeitschrift , 17(1):228–249, 1923.
Vuong Bui, Institut f¨ur Informatik, Freie Universit¨at Berlin, Takustraße 9, 14195 Berlin,Germany
E-mail address : [email protected]@fu-berlin.de