Geodesic
Chromatic Sums for Rooted Planar Triangulations II: The Case λ = τ + 1
Canadian journal of mathematics, Tome 25 (1973) no. 3, pp. 657-671

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In an earlier paper [2] we denned the chromatic sums g, q, l and h. We determined the derivatives of these sums with respect to the colour-numberλ at the special values λ = 1 and λ = 2. In the present paper we find parametric equations for h, l and q in the caseλ = τ + 1, where τ is the golden ratio. We obtain h, l and the basic indeterminate z explicitly in terms of the parameter u, but for q we exhibit only a cubic equation with coefficients depending on u. We obtain an exact formula for the coefficients in h by applying Lagrange's theorem to the parametric equations. 
Tutte, W. T. Chromatic Sums for Rooted Planar Triangulations II: The Case λ = τ + 1. Canadian journal of mathematics, Tome 25 (1973) no. 3, pp. 657-671. doi: 10.4153/CJM-1973-066-8
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[1] 1. Birkhoff, G. D. and Lewis, D. C., Chromatic polynomials, Trans. Amer. Math. Soc. 60 (1946), 355–451. Google Scholar

[2] 2. Tutte, W. T., Chromatic sums for planar triangulations: the cases X = 1 and X = 2, Can. J. Math. 25 (1973), 426–447. Google Scholar

[3] 3. Tutte, W. T., On chromatic polynomials and the golden ratio, J. Combinatorial Theory 9 (1970), 289–296. Google Scholar

[4] 4. Tutte, W. T., The golden ratio in the theory of chromatic polynomials, Annals of the New York Academy of Sciences 175 (1970), 391–402. Google Scholar

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