7 q W ( sq,q ) MC W ( sq,q ) ser. W ( sq,q ) ℓ W ( S$_{2}$$_{F}$ , q ) ℓ W ( S$_{3}$$_{F}$ , q ) ℓ W ( S$_{4}$$_{F}$ , q ) ℓ W ( sc,q ) ser. W ( sc,q ) MC W ( S$_{3}$$_{P}$ , q ) ℓ 4 2.34 2.34 2.33 2.13 2.07 2.04 2.06 1.9 1.78 5 3.25 3.25 3.25 2.96 2.875 2.83 2.75 2.7 2.62 6 4.20 4.20 4.20 3.87 3.765 3.71 3.58 3.6 3.51 7 5.17 5.17 5.17 4.81 4.69 4.64 4.48 − 4.43 8 6.14 6.14 6.14 5.76 5.64 5.58 5.41 − 5.37 9 7.125 7.125 7.125 6.73 6.605 6.54 6.36 − 6.325 10 8.11 8.11 8.11 7.71 7.58 7.51 7.32 − 7.29 100 − 98.0 98.0 97.5 97.4 97.3 97.0 − 97.0 TABLE I: Comparison of lower bounds W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{BC}$$_{z}$ , q ) ℓ for ( L z ) BCz = 2 F = 2 P , 3 F , 4 F , 3 P with approximate values of W (Λ , q ) for the square (sq) and simple cubic (sc) lattices Λ, as de- termined from large- q series expansions, denoted W (Λ , q ) ser. and, where available, Monte Carlo simulations, denoted W (Λ , q ) MC . We also list W ( sq, q ) ℓ for reference. See text for further details. [1] L. Pauling, J. Am. Chem. Soc. 57 , 2680 (1935). [2] W. F. Giauque and J. W. Stout, J. Am. Chem. Soc. 58 1144 (1936). [18] H. Minc, Nonnegative Matrices (New York, Wiley, 1988). [19] D. London, Duke Math. J. 33 , 511 (1966). [20] N. L. Biggs, R. M. Damerell, and D. A. Sands, J. Combin. Theory 12 , 123 (1972). [21] R. Shrock and S.-H. Tsai, Phys. Rev. E 56 , 3935 (1997); J. Phys. A 31 , 9641 (1998); Physica A A265 , 186 (1999). [22] A. Sokal, Combin. Probab. Comput. 18 , 221 (2004). [23] The girth of a graph G , g ( G ) is defined as the number of edges of a minimal-length closed circuit on G . [24] J. F. Nagle, J. Combin. Theory 10 , 42 (1971). [25] G. A. Baker, Jr., J. Combin. Theory 10 , 217 (1971). [3] B. A. Berg, C. Muguruma, and Y. Okamoto, Phys. Rev. B 75 , 092202 (2007). [4] This is in contrast to systems where nonzero ground state entropy is associated with frustration, including the Ising antiferromagnet on the triangular lattice, spin glasses, and spin ice materials such as pyrochlores. [5] F. Y. Wu, Rev. Mod. Phys. 54 , 235 (1982). [6] R. J. Baxter, Exactly Solved Models (Oxford Univ. Press, Oxford, 1982). [26] N. L. Biggs and G. H. J. Meredith, J. Combin. Theory B 20 , 5 (1976). [27] D. Kim and I. G. Enting, J. Combin. Theory B 26 , 327 (1979). [28] A. V. Bakaev and V. I. Kabanovich, J. Phys. A 27 , 6731 (1994). [29] X. Y. Chen and C. Y. Pan, Int. J. Mod. Phys. B 2 , 1495 (1988). [7] Y. Chow and F. Y. Wu, Phys. Rev. B 36 , 285 (1987). [8] For some recent discussions of chromatic polynomials, see talks by N. Biggs, S.-C. Chang, F. M. Dong, B. Jackson, J. Jacobsen, G. Royle, R. Shrock, A. Sokal, C. Thomassen, et al. in Workshop on Zeros of Graph Polynomials (2008), Newton Institute for Mathematical Sciences, Cambridge University, [30] R. Shrock and S.-H. Tsai, Phys. Rev. E 58 , 4332 (1998), cond-mat/9808057. [31] M. Roˇ cek, R. Shrock, and S.-H. Tsai, Physica A 252 , 505 (1998). [32] R. Shrock and S.-H. Tsai, Phys. Rev. E 60 , 3512 (1999); Physica A 275 , 429 (2000). [33] R. Shrock, Phys. Lett. A 261 , 57 (1999). [34] J. Salas and R. Shrock, Phys. Rev. E 64 , 011111 (2001). [35] S.-H. Chang and R. Shrock, Physica A 290 , 402 (2001). [36] N. L. Biggs and R. Shrock, J. Phys. A (Letts) 32 , L489 (1999). [37] E. H. Lieb, Phys. Rev. 162 , 162 (1967). [9] L. Beaudin, J. Ellis-Monaghan, G. Pangborn, and R. Shrock, arXiv:0804.2468. [10] R. Shrock and S.-H. Tsai, Phys. Rev. E 55 , 5165 (1997). [11] N. L. Biggs, Bull. London Math. Soc. 9 , 54 (1977). [12] R. Shrock and S.-H. Tsai, Phys. Rev. E 55 , 6791 (1997). [13] R. Shrock and S.-H. Tsai, Phys. Rev. E 56 , 2733 (1997). [14] R. Shrock and S.-H. Tsai, Phys. Rev. E 56 , 4111 (1997). [15] S.-H. Tsai, Phys. Rev. E 57 , 2686 (1998). [16] S.-C. Chang and R. Shrock, Physica A 316 , 335 (2002). [17] P. Lancaster and M. Tismenetsky, The Theory of Ma- trices, with Applications (New York, Academic Press, 1985). http://www.newton.ac.uk/programmes/CSM/seminars.