Workshop “Zeros of Graph Polynomials”

Chromatic Roots and Fibonacci Numbers

Saeid Alikhani and Yee- hock Peng

Institute for Mathematical Research

University Putra Malaysia

Workshop

“Zeros of Graph Polynomials”

Isaac Newton Institute for Mathematical Science, Cambridge University, UK

21-25 January 2008

Saeid Alikhani, UPM 2

• Outline of Talks

• 1. Introduction• 2. Chromatic roots and golden ratio• 3. Chromatic roots and n-anacci constant

• 4. Some questions


• 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},,2,1{)(: kGVf

)(GEuv)()( vfuf


• 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.


• The number of distinct k-colourings of G, denoted by P(G;k) is called the chromatic polynomial of G.

• A root of P(G;k) 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): (-∞,0), (0,1) , (1,2) and [5,∞) are zero free intervals for all plane triangulations graph.

• Chromatic Zero-free intervals: (-∞,0), (0,1)

• (Jackson 1993): (1,32/27] is also a chromatic zero-free interval.


• (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.


• 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?

*


• 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.

2732,1)1,0()0,(

253

5

B

nk

B k

n

2)( cos4 9,8,7,5n

11n

255

10

B255

10* B


• 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,…


• 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: [Cassini`s Formula]: .

121 FF 21 nnn FFF

))1((51 nn

nF n

n

n FF 1lim

1n

n

FF

1n

n

FF

nnnn FFF )1(2

11


• 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

n

nn FFF

1,0|| 1

1

12

1

n

n

n

n

FF

FF

1

211

n

n

n

n

FF

FF

nF

1

211

n

nnnn F

FFFF


• 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.

)2(1 nFF nnn

rqp 2r

2732

|| rqp

1

11

10

n

nn FFF


• 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.

rqp

rqp * * ]2732,1()1,0()0,(

nn rqp

12 nn FF

p 2nFq 1

1|| nn FFrqp

27321

||1

nF

rqp


• Chromatic Roots and n-anacci Constant • An n-step ( ) Fibonacci sequence :

for k>2

• n-anacci constant:

• It is easy to see that is the real positive root of

• Also note that . (See [8], [14]).

2n

n

i

n

ik

n

k

nn FFFF1

)()()(

2

)(

1 ,1

)(

1

)(

lim n

k

n

k

kn F

F

n

1)( 1 xxxxf nnn

2lim nn


• 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

.

1)( 1 xxxxf nnn

kk

nn

nn aaaGP

11),(

knn aaa ,,, 1 1na nlkalln ,0)1(

0)1,( 1 knn aaaGP


• 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

n2 1)( 1222 xxxxf nnn

)(2 xf n

0)( 2 nP )(|)(2 xPxf n 01)0(2 nf 01)1(2 nf

)(2 xf n )(xP

n2)(2 Nmm

n 1

11),( aaaGP n

nn

n


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 (2n+1)- anacci??

0),( 2 m

nGP

021

)1(

212

m

n

km

nk

mk

n aa

n2 11)( aaQ mmk

kmk

)(|)(2 Qf n 01)0(2 nf 01)1(2 nf)(2 nf )(Q m

),( GP )1,0()0,1( m


• 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 (2n+1)- anacci is a root of

11 ni

ii ain

a

1|| 1

12 n

1)( 21212 xxxxf nnn

n

i

iiaGP

1),(


• Now suppose that there exist a chromatic polynomial , such that, Therefore, there exist , such

that . We have for

• By Theorem 7, , and so

)12(,)( 01 nmaxaxaxP mm 0)( 12 nP

0112

12)( bxbxbxg nmnm

)()()( 12 xgxfxP n 1212 nnm

11212

122

1100

,

,0

bba

bba

baba

nmnm

)(|)1( xPx


• . 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.

0112

12|1 bxbxbx nmnm 012 nma

0||112

11

||11

||

12

1

nm

mm

anmm

mm

amm

a

012 mmnm aaa


• Some questions and remarks

• (Jackson 1993) For any ε>0, there exists a graph G such that

P(G, λ) has a zero in (32/27, 32/27+ ε).

• 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:


• 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.

0),( GP

),( nn BBn

Bn2

cos22

61803.2125 B

987 ,, BBB 10B


• 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]).


• 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,n=2 (or n = 10).

nin iA . }2,1,0{i

0),( GP

),( ,, inin AA }2,1,0{,2 in

2102,2 BA



• 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 Scienti¯c Publishing Co. Pte. Ltd. 2005.


• [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.

1212 xxx nn


• [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-StepNumber.html (last accessed on Dec2006)


• Thanks for your attention!

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Workshop “Zeros of Graph Polynomials”