Chromatic Roots and Fibonacci Numbers Saeid Alikhani and Yee hock Peng Institute for Mathematical Re

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Chromatic Roots and Fibonacci NumbersSaeid
Alikhani and Yee- hock Peng Institute for
Mathematical ResearchUniversity Putra Malaysia

• Workshop
• Zeros of Graph Polynomials
• Isaac Newton Institute for Mathematical Science,
  Cambridge University, UK
• 21-25 January 2008

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• Outline of Talks
• 1. Introduction
• 2. Chromatic roots and golden ratio
• 3. Chromatic roots and n-anacci constant
• 4. Some questions

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• Introduction
• A graph G consists of set V (G) of vertices,
  and
• set E(G) of unordered pairs of vertices
  called edges.
• These graphs are undirected
• A graph is planar if it can be drawn in the plane
  with no edges crossing.
• A (proper) k-colouring of a graph G is a mapping
• ,
  where for every edge

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• The four colour Theorem
• Probably the most famous result in graph theory
  is the following
• theorem
• Four-Colour Theorem
• Every planar graph is 4-colourable.?
• Near-triangulation graphs plane graphs with at
  most one non-triangular face.
• A near- triangulation with 3-face is a
  triangulation.

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• The number of distinct k-colourings of G,
  denoted by P(Gk) is called the chromatic
  polynomial of G.
• A root of P(Gk) is called a chromatic root of
  G.
• An interval is called a root-free interval for a
  chromatic polynomial P(G k) if G has no
  chromatic root in this interval.
• (Birkhoff and Lewis 1946) (-8,0), (0,1) , (1,2)
  and 5,8) are zero free intervals for all plane
  triangulations graph.
• Chromatic Zero-free intervals (-8,0), (0,1)
• (Jackson 1993) (1,32/27 is also a chromatic
  zero-free interval.

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• (Thomassen 1997) There are no more chromatic
  zero-free intervals.
• We recall that a complex number is called an
  algebraic number (resp. an algebraic integer) if
  it is a root of some monic polynomial with
  rational (resp. integer) coefficients.
• Corresponding to any algebraic number ,
  there is a unique monic polynomial p with
  rational coefficients, called the minimal
  polynomial of (over the rationals), with the
  property that p divides every polynomial with
  rational coefficients having as a root.

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• Two algebraic numbers and are called
  conjugate if they have the same minimal
  polynomial.
• Since the chromatic polynomial P(G k) is a monic
  polynomial in k with integer coefficients, its
  roots are, by definition, algebraic integers.
  This naturally raises the question
• Which algebraic integers can occur as roots of
  chromatic polynomials?

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• Clearly those lying in
  are forbidden set.
• Using this reasoning, Tutte 13 proved that
• the Beraha number
  cannot be a chromatic root.
• Salas and Sokal in 10 extended this result
  to show that the
• generalized beraha numbers
  , for
• and , with k coprime to n, are
  never chromatic roots. For n
• 10 they showed the weaker result that
  and
• are not chromatic roots of any plane
  near-triangulation.

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• Fibonacci numbers are terms of the sequence
  defined in a quite simple recursive fashion.
• However, despite its simplicity, they have some
  curious properties which are worth attention.
• 1,1,2,3,5,8,13,21,

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• Golden Ratio and Chromatic Roots
• Fibonacci sequence and
  
• Golden ratio
• Theorem1 For every natural number n,
  .?
• Corollary 1 If n is even and
  if n is odd .?
• Theorem 2 Cassinis Formula
  .?

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• Theorem 3 For every natural n,
• Proof. Suppose that n is even, therefore n-1
  is odd, and by
• Corollary 1, we have
  ,and hence
• and by multiplying in this inequality,
  we have
• 
  

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• Thus by Theorem 2, we have
  . Similarly, the result holds
  when n is odd.?
• Theorem 4.( 7, P.78)
  .?
• The following theorem is a consequence of
  Salas-Sokal Proposition in 10
  
• Theorem 5. Consider a number of the form
  , where p, q are rational,
  is an integer that is not a
• perfect square, and
  . Then is not the root of any
  chromatic polynomial.

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• Proof. If is a root of
  some polynomial with integer coeffcients (e.g. a
  chromatic polynomial), then so its conjugate
• .But or can
  lie in
• a contradiction.?
• Corollary 2 For every natural n, cannot
  be a root of any chromatic polynomials.
• Proof. By Theorem 4, we can consider
  of the form
• with and
  . Since
  
• by Theorem 3,
  .Therefore we have the result by Theorem
  5.?

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• Chromatic Roots and n-anacci Constant
• An n-step ( ) Fibonacci sequence
• for kgt2
• n-anacci constant
• It is easy to see that is the real
  positive root of
• 
  
• Also note that . (See 8,
  14).

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• Theorem 6. (8) The polynomial
  is an
  irreducible polynomial over Q. ?
• Theorem 7. (4) Let G be a graph with n vertices
  and k connected components. Then the chromatic
  polynomial of G is of the form
• with integer,
  , and
  Furthermore, if G has at least one edge, then
• 
  .?

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• Theorem 8. For every natural n, the 2n-anacci
  numbers cannot be roots of any chromatic
  polynomials.
• Proof. We know that is a root of
  
• which is minimal polynomial for this root.
  It is obvious that is not a chromatic
  polynomial. Now suppose that there exist a
  chromatic polynomial P(x) such that
  . By Theorem 6,
  . Since , and
• by the intermediate value theorem,
  and therefore has a root in (-1, 0)
  and this is a contradiction.?
• Theorem 9. All natural powers of cannot
  be chromatic root.
• Proof. Suppose that is a
  chromatic root, that is there exist a chromatic
  polynomial

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• Such that .
  Therefore
  
• So we can say that is a root of the
  polynomial,
• so . Since
  and
• and so have a root say
  in (-1,0). Therefore
• is a root of . Since
  , we have a
  contradiction.?
• How about (2n1)- anacci??

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• We think that (2n 1)-anacci numbers and all
  natural power of them also cannot be chromatic
  roots, but we are not able to prove it!
• Theorem 10. (Dong et al 4) Let
  be a chromatic polynomial of a
  graph G of order n. Then for any
  
• (where
  equality holds if and only if G is a tree).?
• Theorem 11. For every natural n, can not
  be a root of chromatic polynomial of graph G with
  at most 4n 2 vertices.
• Proof. We know that (2n1)- anacci is a root of

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• Now suppose that there exist a chromatic
  polynomial
• 
  , such that,
  
• Therefore, there exist
  , such
• that .
  We have for
• By Theorem 7, , and so

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• . By the
  above equalities, we have .
  By Theorem 10,
• So, we have
  . Therefore P(x) cannot be a chromatic
  polynomial, and this is a contradiction.?

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• Some questions and remarks
• (Jackson 1993) For any egt0, there exists a graph
  G such that
• P(G, ?) has a zero in (32/27, 32/27 e).?
• Theorem above says what is on the right of the
  number of 32/27 in general case. But the problem
  has been considered for some families of graphs
  as well. One of this families is triangulation
  graphs, and there are some open problems for it.
  We recall the Beraha question, which says

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• Question 1. (Beraha's question 1) Is it true
  for every , there exists a plane
  triangulation G such that has a
  root in,
• , where
  is called the n-th Beraha
  constant (or number)?
• Beraha et al. 3 proved that
• is an accumulation point of real chromatic
  roots of certain plane triangulations.
• Jacobsen et al. 6 extended this to show that
  and are likewise
  accumulation points of real chromatic roots of
  plane triangulations.

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• Finally, Royle 9 has recently exhibited a
  family of plane triangulations with chromatic
  roots converging to 4.
• Of course, it is an open question which other
  numbers in the interval
• ( 32/27, 4) can be accumulation points of
  real chromatic roots of planar graphs.
• The following conjecture of Thomassen is one
  possible answer.
• Conjecture 1. The set of chromatic roots of the
  family of planar graphs consists of 0,1 and a
  dense subset of ( 32/27,4).(See 12).

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• Now, let , where
  .
• Here we ask the following question that is
  analogous to Beraha's question.
• Question 2. Is it true that, for any ,
  there exists a plane triangulation graph G such
  that has a root in
• ?(
  ).?
• Note that .
  Beraha, Kahane and Reid 2 proved that the
  answer to our question (or respectively, Beraha
  question) is positive for i,n2 (or n 10).

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• References
• 1 Beraha, Infinite non-interval families of
  maps and chromials, Ph.D. thesis, Johns Hopkins
  University, 1975.
• 2 Beraha,S., Kahane, J, and R. Reid, B7 and B10
  are limit points of chromatic zeros, Notices
  Amer. Math. Soc. 20(1973), 45.
• 3 Beraha,S., Kahane,J. and Weiss, N.J., Limits
  of chromatic zeros of some families of maps, J.
  Combinatorial Theory Ser. B 28 (1980), 52-65.
• 4 Dong, F.M, Koh, K. M, Teo, K. L, Chromatic
  polynomial and chromaticity of graphs, World
  Scientic Publishing Co. Pte. Ltd. 2005.

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• 5 Jackson, B., A zero free interval for
  chromatic polynomials of graphs, Combin.Probab.
  Comput. 2 (1993) 325-336.
• 6 J.L. Jacobsen, J. Salas and A.D. Sokal,
  Transfer matrices and partition-function zeros
  for antiferromagnetic Potts models. III.
  Triangular-lattice chromatic polynomial,J.Statist.
  Phys. 112 (2003), 921-1017, see e.g. Tables 3
  and 4.
• 7 Koshy, T., Fibonacci and Lucas numbers with
  applications, A Willey-Interscience Publication,
  2001.
• 8 Martin, P. A, The Galois group of
  , Journal of pure and
  applied algebra. 190 (2004) 213-223.
• 9 G. Royle, Planar triangulations with real
  chromatic roots arbitrarily close to four,
  http//arxiv.org/abs/math.CO/0511304.

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• 10 J. Salas and A.D. Sokal, Transfer matrices
  and partition-function zeros for
  antiferromagnetic Potts models. I. General theory
  and square-lattice chromatic polynomial,J.
  Statist. Phys. 104 (2001), 609-699.
• 11 I. Stewart and D. Tall, Algebraic Number
  Theory, 2nd ed, Chapman and Hall,London- New
  York, 1987.
• 12 Thomassen, C, The zero- free intervals for
  chromatic polynomials of graphs, Combin.Probab.
  Comput. 6,(1997), 497-506.
• 13 Tutte, W.T., On chromatic polynomials and
  golden ratio, J. Combinatorial Theory,Ser B 9
  (1970),289-296.
• 14 http//mathworld.wolfram.com/Finonaccin-StepN
  umber.html (last accessed on Dec2006)

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• Thanks for your attention!