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Complex Roots of Chromatic Polynomials Norman Biggs INI

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Complex Roots of Chromatic Polynomials Norman Biggs INI Complex Roots of Chromatic Polynomials Norman Biggs INI Workshop January 2008 1 1. Preview The original stimulus was the work of Tutte (1970) concerning the real roots of chromatic polynomials of planar graphs. Biggs, Damerell and Sands (1972) studied complex roots of recursive families, such as the cyclic ladders Ln: They found that the chromatic polynomial of Ln is (z2 − 3z + 3)n + (z − 1)(3 − z)n + (z − 1)(1 − z)n + (z2 − 3z + 1). 2 They also observed that as n → ∞ the roots cluster around certain curves. At the time, the form of the polynomial and the behaviour of the roots was unexplained. 3 A theorem of Beraha, Kahane and Weiss (1980) was used by Read and Royle (1991) to explain the behaviour of the roots. The polynomial (z2 − 3z + 3)n + (z − 1)(3 − z)n + (z − 1)(1 − z)n + (z2 − 3z + 1) has the form X n µi(z)λi(z) . The limit curves are (parts of) the equimodular curves, defined by |λi(z)| = |λj(z)| (i 6= j). In this case |z2−3z+3| = |z−3|, |z2−3z+3| = |z−1|, |z−3| = |z−1|. 4 In 1999 this idea was revisited. A bracelet is constructed by taking n copies of a graph B and linking each copy to the next (in cyclic order) with a set of links L. 5 It can be shown that the number of k-colourings of a bracelet can be expressed in the form trace T (k)n, where the transfer matrix T (k) is a (0, 1)-matrix whose size is a function of k. Replacing k by z ∈ C it follows that the chromatic polynomials of bracelets have the form n X n trace T (z) = µi(z)λi(z) , where the λi are the eigenvalues of T and the µi are their multiplicities. The Beraha-Kahane-Weiss theorem applies. This means that we must find the equimodular curves associated with the eigenvalues of matrices of the form T (z): that is, curves where T (z) has two eigenvalues with equal modulus. 6 In some cases it is easy to find the irreducible constituents of T (z) and their eigenvalues. Example 1 For the ladder graphs there are three constituents ! 2 2 − z 1 T0(z) = (z − 3z + 3),T1(z) = ,T2(z) = (1), 1 2 − z with multiplicities 1, z − 1, z2 − 3z + 1. The chromatic polynomial is therefore 1(z2 − 3z + 3)n + (z − 1)((3 − z)n + (1 − z)n) + (z2 − 3z + 1)1n. 7 Biggs, Reinfeld, and Chang studied the generalised dodecahedra, where the components are paths of length 4. The chromatic polynomial of Dn can be written as n n 2 n tr(T0(z) ) + (z − 1)tr(T1(z) ) + (z − 3z + 1)tr(T2(z) ) +(z3 − 5z2 + 6z − 1). 8 Here the constituents T0, T1, T2 are small matrices (size 3,6,4) with polynomial entries. Their eigenvalues are not polynomials, but it can be shown that the limit curves for Dn are of the following form (roughly!). We shall explain how the curves are made up, including the reason for the apparent ‘triple points’ . 9 There are several things to do. 2. Formalize the algebra of bracelet colourings. 3. Explain how to compute the equimodular curves, and hence the limit curves. 4. Study the topology of the curves. 5. 6. 10 2: Algebra of bracelet colourings This is a brief survey. More details are given in: Chromatic polynomials and representations of the symmetric group Linear Algebra and Appl. 356 (2002) 3-26. Specht modules and chromatic polynomials J. Combinatorial Theory (B) 92 (2004) 359-377. 11 We begin with the transfer matrix for bracelets. For simplicity, we shall take the components B to be complete graphs Kb. Two k-colourings α, β of Kb are L-compatible if α(v) 6= β(w), for all (v, w) ∈ L. This means that the graph formed by two copies of Kb linked by L is properly coloured by colouring the Kb’s as in α and β. Define the transfer matrix TL by 1 if α and β are L-compatible; (T ) = L αβ 0 otherwise. So, for each k, TL is a (0, 1)-matrix whose size is k(k − 1) ··· (k − b + 1). 12 Theorem The number of k-colourings of a bracelet with n n components is the trace of (TL) . Proof Write T = TL. Then n X n X X trace T = (T )αα = TαβTβγ ··· Tωα. α α β,γ,...,ω Now TαβTβγ ··· Tωα is nonzero (and equal to 1) if and only if the colourings α, β, γ, . , ω unite to form a proper colouring of the whole bracelet. Because the names of the k colours are irrelevant, it turns out that T can be represented by constituent matrices Ts(k) with fixed size, but with entries and multiplicities that are polynomial functions of k. 13 For bracelets with Kb components we can give general rules. For any k ≥ b, a permutation ω ∈ Symk acts on the k-colourings of Kb in the obvious way: α 7→ ωα. There is a corresponding matrix R(ω) operating on the space of k-colourings: 1 if β = ωα; R(ω) = αβ 0 otherwise. Lemma For any ω ∈ Symk, TLR(ω) = R(ω)TL. 14 The reduction of R(ω) to its irreducible constituents can be expressed in the standard theory of representations of the symmetric group (Specht modules). Essentially, Symk is acting on the set of injections of a b-set into a k-set. Another standard result of representation theory (Schur’s Lemma) gives the relationship between the constituents of TL and those of R(ω). The outcome is a simple description of the multiplicity and size of the constituents TL,π of TL, corresponding to integer partitions π . 15 Notation When π is a partition of the integer ` we write |π| = ` and denote the parts by π1 ≥ π2 ≥ · · · ≥ π`, where any part may be zero. Put σi = πi + ` − i and define `! × Q(σ − σ ) = i j dπ Q . σi! Each π such that |π| = ` determines an irreducible π representation S of the symmetric group Sym`, with dimension dπ. These form a complete set. For example, when ` = 3 the irreducible representations are S[300] S[210] S[111] with dimensions 1 2 1. 16 Theorem The number of k-colourings of a bracelet Bn with components Kb and links L can be written as a sum of terms, one for each partition π with 0 ≤ |π| ≤ b: X n mπ(k) trace(TL,π) . π Here mπ(k) = fπ(k − σ1)(k − σ2) ··· (k − σ|π|), where fπ = dπ/|π|!, σi = |π| + πi − i, and TL,π is a square matrix in which the entries are polynomials b in k that depend on L, and the size is |π| dπ, independent of L. 17 Example Suppose the components are K3’s and the links join corresponding vertices. 18 Here b = 3 and the multiplicities are: m[0](k) = 1 m[1](k) = k − 1 1 1 m (k) = (k − 3)k m (k) = (k − 2)(k − 1) [20] 2 [11] 2 1 m (k) = (k − 5)(k − 1)k [300] 6 2 m (k) = (k − 4)(k − 2)k [210] 6 1 m (k) = (k − 3)(k − 2)(k − 1). [111] 6 Replacing k by z, and calculating the eigenvalues of the matrices TL,π when L is the given set of links, we get the chromatic polynomial originally found by Biggs and Shrock 1999: 19 π = [0] : 1(z3 − 6z2 + 14z − 13)n π = [1] : +(z − 1){(−z2 + 7z − 13)n + 2(−z2 + 4z − 4)n} 1 π = [20] : + (z2 − 3z){(z − 5)n + 2(z − 2)n} 2 1 π = [11] : + (z2 − 3z + 2){2(z−4)n+(z−1)n} 2 π = [300], [210], [111] : +(z3 − 6z2 + 8z − 1)(−1)n. In the last line the multiplicity is the sum of the terms corresponding to the partitions of 3. In this example the eigenvalues of the constituents are polynomials. The eigenvalues depend on the links, but the multiplicities would be the same for any bracelet with K3 components. 20 When the components of the bracelet are not complete, we look at each class of k-colourings separately, and ‘collapse’ accordingly. For example, when the components are paths of length 4, (as for the generalized dodecahedra) there are five classes: 21 3: Equimodular curves and limit curves For more details, see: Equimodular curves Discrete Mathematics 259 (2002) 37-57. Also three research reports: LSE-CDAM-2000-07, LSE-CDAM-2000-17, LSE-CDAM-2001-01 available via the LSE Mathematics Department website. 22 The Story So Far • The chromatic polynomial of a bracelet with n components can be written in the form X n µi(z)λi(z) , where the λi(z) are eigenvalues of certain matrices. • As n → ∞ the roots of such a family of polynomials approach parts of the equimodular curves defined by |λi(z)| = |λj(z)|. We must explain what is meant by parts of. 23 Let us say that an eigenvalue λm(z) is dominant at z0 if |λm(z0)| ≥ |λi(z0)| for all eigenvalues λi(z). Consider an equimodular curve |λa(z)| = |λb(z)|. If a part of the curve is the boundary between a region in which λa(z) is dominant and a region in which λb(z) is dominant, we shall say that this is a dominant part of the curve. The Beraha-Kahane-Weiss theorem says that only dominant parts of the equimodular curves can be limit points of zeros.
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