-
1
-
-
0000331178
-
The third law of thermodynamics and the degeneracy of the ground state for lattice systems
-
M. Aizenman, E.H. Lieb, The third law of thermodynamics and the degeneracy of the ground state for lattice systems, J. Stat. Phys. 24 (1981) 279-297; Y. Chow, F.Y. Wu, Residual entropy and the validity of the third law of thermodynamics in discrete spin systems, Phys. Rev. B 36 (1987) 285-288.
-
(1981)
J. Stat. Phys.
, vol.24
, pp. 279-297
-
-
Aizenman, M.1
Lieb, E.H.2
-
2
-
-
0000228409
-
Residual entropy and the validity of the third law of thermodynamics in discrete spin systems
-
M. Aizenman, E.H. Lieb, The third law of thermodynamics and the degeneracy of the ground state for lattice systems, J. Stat. Phys. 24 (1981) 279-297; Y. Chow, F.Y. Wu, Residual entropy and the validity of the third law of thermodynamics in discrete spin systems, Phys. Rev. B 36 (1987) 285-288.
-
(1987)
Phys. Rev. B
, vol.36
, pp. 285-288
-
-
Chow, Y.1
Wu, F.Y.2
-
3
-
-
36149036355
-
Chromatic zeros of large planar lattices
-
R.J. Baxter, Chromatic zeros of large planar lattices, J. Phys. A 20 (1987) 5241-5261.
-
(1987)
J. Phys. A
, vol.20
, pp. 5241-5261
-
-
Baxter, R.J.1
-
4
-
-
33746898508
-
Is the four-color conjecture almost false?
-
S. Beraha, J. Kahane, N.N. Weiss, Is the four-color conjecture almost false?, J. Combin. Theory B 27 (1979) 1-12; Limits of chromatic zeros for some families of graphs, J. Combin. Theory B 28 (1980) 52-65.
-
(1979)
J. Combin. Theory B
, vol.27
, pp. 1-12
-
-
Beraha, S.1
Kahane, J.2
Weiss, N.N.3
-
5
-
-
0001870053
-
Limits of chromatic zeros for some families of graphs
-
S. Beraha, J. Kahane, N.N. Weiss, Is the four-color conjecture almost false?, J. Combin. Theory B 27 (1979) 1-12; Limits of chromatic zeros for some families of graphs, J. Combin. Theory B 28 (1980) 52-65.
-
(1980)
J. Combin. Theory B
, vol.28
, pp. 52-65
-
-
-
6
-
-
0001844357
-
Colouring square lattice graphs
-
N.L. Biggs, Colouring square lattice graphs, Bull. London Math. Soc. 9 (1977) 54-56.
-
(1977)
Bull. London Math. Soc.
, vol.9
, pp. 54-56
-
-
Biggs, N.L.1
-
10
-
-
4243154505
-
Approximations for chromatic polynomials
-
N.L. Biggs, G.H. Meredith, Approximations for chromatic polynomials, J. Combin. Theory B 20 (1976) 5-19.
-
(1976)
J. Combin. Theory B
, vol.20
, pp. 5-19
-
-
Biggs, N.L.1
Meredith, G.H.2
-
11
-
-
0033584757
-
T = 0 partition functions for Potts antiferromagnets on square lattice strips with (twisted) periodic boundary conditions
-
N.L. Biggs, R. Shrock, T = 0 partition functions for Potts antiferromagnets on square lattice strips with (twisted) periodic boundary conditions, J. Phys. Lett. A 32 (1999) L489-L493.
-
(1999)
J. Phys. Lett. A
, vol.32
-
-
Biggs, N.L.1
Shrock, R.2
-
12
-
-
0002741878
-
A determinant formula for the number of ways of coloring a map
-
G.D. Birkhoff, A determinant formula for the number of ways of coloring a map, Ann. Math. 14 (1912) 42-46.
-
(1912)
Ann. Math.
, vol.14
, pp. 42-46
-
-
Birkhoff, G.D.1
-
13
-
-
0007161964
-
Chromatic roots - Some observations and conjectures
-
E.J. Farrell, Chromatic roots - some observations and conjectures, Discrete Math. 29 (1980) 161-167.
-
(1980)
Discrete Math.
, vol.29
, pp. 161-167
-
-
Farrell, E.J.1
-
14
-
-
0002862954
-
Phase transitions in lattice systems with random local proerties
-
P.W. Kasteleyn, C.M. Fortuin, Phase transitions in lattice systems with random local proerties, J. Phys. Soc. Japan 26 (Suppl.) (1969) 11-14; C.M. Fortuin, P.W. Kasteleyn, On the random cluster model. I. Introduction and relation to other models, Physica 57 (1972) 536-564.
-
(1969)
J. Phys. Soc. Japan
, vol.26
, Issue.SUPPL.
, pp. 11-14
-
-
Kasteleyn, P.W.1
Fortuin, C.M.2
-
15
-
-
0041855748
-
On the random cluster model. I. Introduction and relation to other models
-
P.W. Kasteleyn, C.M. Fortuin, Phase transitions in lattice systems with random local proerties, J. Phys. Soc. Japan 26 (Suppl.) (1969) 11-14; C.M. Fortuin, P.W. Kasteleyn, On the random cluster model. I. Introduction and relation to other models, Physica 57 (1972) 536-564.
-
(1972)
Physica
, vol.57
, pp. 536-564
-
-
Fortuin, C.M.1
Kasteleyn, P.W.2
-
16
-
-
36049056500
-
Residual entropy of square ice
-
E.H. Lieb, Residual entropy of square ice, Phys. Rev. 162 (1967) 162-172.
-
(1967)
Phys. Rev.
, vol.162
, pp. 162-172
-
-
Lieb, E.H.1
-
17
-
-
12244272075
-
A connection between complex-temperature properties of the 1D and 2D spin s Ising model
-
V. Matveev, R. Shrock, A connection between complex-temperature properties of the 1D and 2D spin s Ising model, Phys. Lett. A 204 (1995) 353-358.
-
(1995)
Phys. Lett. A
, vol.204
, pp. 353-358
-
-
Matveev, V.1
Shrock, R.2
-
18
-
-
0004808372
-
Coefficients of chromatic polynomials
-
G.H. Meredith, Coefficients of chromatic polynomials, J. Combin. Theory B 13 (1972) 14-17.
-
(1972)
J. Combin. Theory B
, vol.13
, pp. 14-17
-
-
Meredith, G.H.1
-
19
-
-
0001860452
-
A new subgraph expansion for obtaining coloring polynomials for graphs
-
J.F. Nagle, A new subgraph expansion for obtaining coloring polynomials for graphs, J. Combin. Theory 10 (1971) 42-59; G.A. Baker Jr., Linked-cluster expansion for the graph-vertex coloration problem, J. Combin. Theory 10 (1971) 217-231.
-
(1971)
J. Combin. Theory
, vol.10
, pp. 42-59
-
-
Nagle, J.F.1
-
20
-
-
0007761339
-
Linked-cluster expansion for the graph-vertex coloration problem
-
J.F. Nagle, A new subgraph expansion for obtaining coloring polynomials for graphs, J. Combin. Theory 10 (1971) 42-59; G.A. Baker Jr., Linked-cluster expansion for the graph-vertex coloration problem, J. Combin. Theory 10 (1971) 217-231.
-
(1971)
J. Combin. Theory
, vol.10
, pp. 217-231
-
-
Baker Jr., G.A.1
-
21
-
-
84926736518
-
Some generalized order-disorder transformations
-
R.B. Potts, Some generalized order-disorder transformations, Proc. Cambridge Philos Soc. 48 (1952) 106-109.
-
(1952)
Proc. Cambridge Philos Soc.
, vol.48
, pp. 106-109
-
-
Potts, R.B.1
-
22
-
-
0002529522
-
An introduction to chromatic polynomials
-
R.C. Read, An introduction to chromatic polynomials, J. Combin. Theory 4 (1968) 52-71.
-
(1968)
J. Combin. Theory
, vol.4
, pp. 52-71
-
-
Read, R.C.1
-
23
-
-
0011641770
-
A large family of chromatic polynomials
-
R.C. Read, A large family of chromatic polynomials, in: Proceedings of the Third Caribbean Conference on Combinatorics and Computing, 1981, pp. 23-41; Recent advances in chromatic polynomial theory, in: Proceedings of the Fifth Caribbean Conference on Combinatorics and Computing, 1988, pp. 1-13.
-
(1981)
Proceedings of the Third Caribbean Conference on Combinatorics and Computing
, pp. 23-41
-
-
Read, R.C.1
-
24
-
-
0347286608
-
Recent advances in chromatic polynomial theory
-
R.C. Read, A large family of chromatic polynomials, in: Proceedings of the Third Caribbean Conference on Combinatorics and Computing, 1981, pp. 23-41; Recent advances in chromatic polynomial theory, in: Proceedings of the Fifth Caribbean Conference on Combinatorics and Computing, 1988, pp. 1-13.
-
(1988)
Proceedings of the Fifth Caribbean Conference on Combinatorics and Computing
, pp. 1-13
-
-
-
25
-
-
0001358201
-
Chromatic roots of families of graphs
-
Wiley, New York
-
R.C. Read, G.F. Royle, Chromatic roots of families of graphs, Graph Theory, Combinatorics, and Applications, Vol. 2, Wiley, New York, 1991, pp. 1009-1029.
-
(1991)
Graph Theory, Combinatorics, and Applications
, vol.2
, pp. 1009-1029
-
-
Read, R.C.1
Royle, G.F.2
-
26
-
-
0002461816
-
Chromatic polynomials
-
Academic Press, New York
-
R.C. Read, W.T. Tutte, Chromatic polynomials, Selected Topics in Graph Theory, 3, Academic Press, New York, 1988, pp. 15-42.
-
(1988)
Selected Topics in Graph Theory
, vol.3
, pp. 15-42
-
-
Read, R.C.1
Tutte, W.T.2
-
27
-
-
0032047241
-
Chromatic polynomials for families of strip graphs and their asymptotic limits
-
M. Roček, R. Shrock, S.-H. Tsai, Chromatic polynomials for families of strip graphs and their asymptotic limits, Physica A 252 (1998) 505-546.
-
(1998)
Physica A
, vol.252
, pp. 505-546
-
-
Roček, M.1
Shrock, R.2
Tsai, S.-H.3
-
28
-
-
0032183844
-
Chromatic polynomials on J(Π H)I strip graphs and their asymptotic limits
-
M. Roček, R. Shrock, S.-H. Tsai, Chromatic polynomials on J(Π H)I strip graphs and their asymptotic limits, Physica A 259 (1998) 367-387.
-
(1998)
Physica A
, vol.259
, pp. 367-387
-
-
Roček, M.1
Shrock, R.2
Tsai, S.-H.3
-
29
-
-
0004287752
-
-
Springer, New York
-
I.R. Shafarevich, Basic Algebraic Geometry, Springer, New York, 1974; R. Hartshorne, Algebraic Geometry, Springer, New York, 1977.
-
(1974)
Basic Algebraic Geometry
-
-
Shafarevich, I.R.1
-
30
-
-
0003897060
-
-
Springer, New York
-
I.R. Shafarevich, Basic Algebraic Geometry, Springer, New York, 1974; R. Hartshorne, Algebraic Geometry, Springer, New York, 1977.
-
(1977)
Algebraic Geometry
-
-
Hartshorne, R.1
-
31
-
-
0346704344
-
T = 0 partition functions for Potts antiferromagnets on Möbius strips and effects of graph topology
-
R. Shrock, T = 0 partition functions for Potts antiferromagnets on Möbius strips and effects of graph topology, Phys. Lett. A 261 (1999) 57-62.
-
(1999)
Phys. Lett. A
, vol.261
, pp. 57-62
-
-
Shrock, R.1
-
32
-
-
0034250847
-
Exact Potts model partition functions for ladder graphs
-
R. Shrock, Exact Potts model partition functions for ladder graphs, Physica A 283 (2000) 388-446.
-
(2000)
Physica A
, vol.283
, pp. 388-446
-
-
Shrock, R.1
-
33
-
-
0000240640
-
Ground state entropy of Potts antiferromagnets and the approach to the 2D thermodynamic limit
-
cond-mat/9808057
-
R. Shrock, S.-H. Tsai, Ground state entropy of Potts antiferromagnets and the approach to the 2D thermodynamic limit, Phys. Rev. E 58 (1988) 4332-4339 (cond-mat/9808057).
-
(1988)
Phys. Rev. E
, vol.58
, pp. 4332-4339
-
-
Shrock, R.1
Tsai, S.-H.2
-
34
-
-
0000931941
-
Asymptotic limits and zeros of chromatic polynomials and ground state entropy of Potts antiferromagnets
-
R. Shrock, S.-H. Tsai, Asymptotic limits and zeros of chromatic polynomials and ground state entropy of Potts antiferromagnets, Phys. Rev. E 55 (1997) 5165-5179.
-
(1997)
Phys. Rev. E
, vol.55
, pp. 5165-5179
-
-
Shrock, R.1
Tsai, S.-H.2
-
35
-
-
0001223998
-
Families of graphs with chromatic zeros lying on circles
-
R. Shrock, S.-H. Tsai, Families of graphs with chromatic zeros lying on circles, Phys. Rev. E 56 (1997) 1342-1345.
-
(1997)
Phys. Rev. E
, vol.56
, pp. 1342-1345
-
-
Shrock, R.1
Tsai, S.-H.2
-
36
-
-
0000107005
-
r({G}, q) functions that are non-analytic at 1/q = 0
-
r({G}, q) functions that are non-analytic at 1/q = 0, Phys. Rev. E 56 (1997) 3935-3943.
-
(1997)
Phys. Rev. E
, vol.56
, pp. 3935-3943
-
-
Shrock, R.1
Tsai, S.-H.2
-
37
-
-
0001679220
-
Lower bounds and series for the ground state entropy of the Potts antiferromagnet on Archimedean lattices and their duals
-
R. Shrock, S.-H. Tsai, Lower bounds and series for the ground state entropy of the Potts antiferromagnet on Archimedean lattices and their duals, Phys. Rev. E 56 (1997) 4111-4124.
-
(1997)
Phys. Rev. E
, vol.56
, pp. 4111-4124
-
-
Shrock, R.1
Tsai, S.-H.2
-
38
-
-
0000647605
-
Upper and lower bounds for ground state entropy of antiferromagnetic Potts models
-
R. Shrock, S.-H. Tsai, Upper and lower bounds for ground state entropy of antiferromagnetic Potts models, Phys. Rev. E 55 (1997) 6791-6794; Ground state entropy of Potts antiferromagnets: bounds, series, and Monte Carlo measurements, Phys. Rev. E 56 (1997) 2733-2737.
-
(1997)
Phys. Rev. E
, vol.55
, pp. 6791-6794
-
-
Shrock, R.1
Tsai, S.-H.2
-
39
-
-
4243808666
-
Ground state entropy of Potts antiferromagnets: Bounds, series, and Monte Carlo measurements
-
R. Shrock, S.-H. Tsai, Upper and lower bounds for ground state entropy of antiferromagnetic Potts models, Phys. Rev. E 55 (1997) 6791-6794; Ground state entropy of Potts antiferromagnets: bounds, series, and Monte Carlo measurements, Phys. Rev. E 56 (1997) 2733-2737.
-
(1997)
Phys. Rev. E
, vol.56
, pp. 2733-2737
-
-
-
40
-
-
0032183370
-
Ground state entropy of Potts antiferromagnets on homeomorphic families of strip graphs
-
R. Shrock, S.-H. Tsai, Ground state entropy of Potts antiferromagnets on homeomorphic families of strip graphs, Physica A 259 (1998) 315-348.
-
(1998)
Physica A
, vol.259
, pp. 315-348
-
-
Shrock, R.1
Tsai, S.-H.2
-
41
-
-
0033617045
-
Ground state entropy of the Potts antiferromagnet on cyclic strip graphs
-
R. Shrock, S.-H. Tsai, Ground state entropy of the Potts antiferromagnet on cyclic strip graphs, J. Phys. Lett. A 32 (1999) L195-L200.
-
(1999)
J. Phys. Lett. A
, vol.32
-
-
Shrock, R.1
Tsai, S.-H.2
-
42
-
-
0032483925
-
Ground state degeneracy of Potts antiferromagnets: Cases with noncompact W boundaries having multiple points at 1/q = 0
-
R. Shrock, S.-H. Tsai, Ground state degeneracy of Potts antiferromagnets: cases with noncompact W boundaries having multiple points at 1/q = 0, J. Phys. A 31 (1998) 9641-9665.
-
(1998)
J. Phys. A
, vol.31
, pp. 9641-9665
-
-
Shrock, R.1
Tsai, S.-H.2
-
43
-
-
0033538388
-
Ground state entropy of Potts antiferromagnets on cyclic polygon chain graphs
-
R. Shrock, S.-H. Tsai, Ground state entropy of Potts antiferromagnets on cyclic polygon chain graphs, J. Phys. A 32 (1999) 5053-5070.
-
(1999)
J. Phys. A
, vol.32
, pp. 5053-5070
-
-
Shrock, R.1
Tsai, S.-H.2
-
44
-
-
0033101214
-
Ground state degeneracy of Potts antiferromagnets: Homeomorphic classes with noncompact W boundaries
-
R. Shrock, S.-H. Tsai, Ground state degeneracy of Potts antiferromagnets: homeomorphic classes with noncompact W boundaries, Physica A 265 (1999) 186-223.
-
(1999)
Physica A
, vol.265
, pp. 186-223
-
-
Shrock, R.1
Tsai, S.-H.2
-
45
-
-
0000793230
-
Ground state degeneracy of Potts antiferromagnets on 2D lattices: Approach using infinite cyclic strip graphs
-
R. Shrock, S.-H. Tsai, Ground state degeneracy of Potts antiferromagnets on 2D lattices: approach using infinite cyclic strip graphs, Phys. Rev. E 60 (1999) 3512-3515; Exact partition functions for Potts antiferromagnets on cyclic lattice strips, Physica A 275 (2000) 429-449.
-
(1999)
Phys. Rev. E
, vol.60
, pp. 3512-3515
-
-
Shrock, R.1
Tsai, S.-H.2
-
46
-
-
0033901002
-
Exact partition functions for Potts antiferromagnets on cyclic lattice strips
-
R. Shrock, S.-H. Tsai, Ground state degeneracy of Potts antiferromagnets on 2D lattices: approach using infinite cyclic strip graphs, Phys. Rev. E 60 (1999) 3512-3515; Exact partition functions for Potts antiferromagnets on cyclic lattice strips, Physica A 275 (2000) 429-449.
-
(2000)
Physica A
, vol.275
, pp. 429-449
-
-
-
47
-
-
85086351888
-
Bounds on zeros of (di)chromatic polynomials and Potts model partition functions
-
in press
-
A. Sokal, Bounds on zeros of (di)chromatic polynomials and Potts model partition functions, Combin. Probab. Comput., in press.
-
Combin. Probab. Comput.
-
-
Sokal, A.1
-
48
-
-
0002092950
-
A contribution to the theory of chromatic polynomials
-
W.T. Tutte, A contribution to the theory of chromatic polynomials, Canad. J. Math. 6 (1954) 80-91.
-
(1954)
Canad. J. Math.
, vol.6
, pp. 80-91
-
-
Tutte, W.T.1
-
49
-
-
0000537710
-
On dichromatic polynomials
-
W.T. Tutte, On dichromatic polynomials, J. Combin. Theory 2 (1967) 301-320.
-
(1967)
J. Combin. Theory
, vol.2
, pp. 301-320
-
-
Tutte, W.T.1
-
50
-
-
0003354205
-
Graph Theory
-
G.C. Rota (Ed.), Addison-Wesley, New York
-
W.T. Tutte, Graph Theory, in: G.C. Rota (Ed.), Encyclopedia of Mathematics and its Applications, Vol. 21, Addison-Wesley, New York, 1984.
-
(1984)
Encyclopedia of Mathematics and Its Applications
, vol.21
-
-
Tutte, W.T.1
-
52
-
-
0000217456
-
The coloring of graphs
-
H. Whitney, The coloring of graphs, Ann. Math. 33 (1932) 688-718; A logical expansion in mathematics, Bull. Amer. Math. Soc. 38 (1932) 572-579.
-
(1932)
Ann. Math.
, vol.33
, pp. 688-718
-
-
Whitney, H.1
-
53
-
-
6244228084
-
A logical expansion in mathematics
-
H. Whitney, The coloring of graphs, Ann. Math. 33 (1932) 688-718; A logical expansion in mathematics, Bull. Amer. Math. Soc. 38 (1932) 572-579.
-
(1932)
Bull. Amer. Math. Soc.
, vol.38
, pp. 572-579
-
-
-
54
-
-
0042492825
-
The Potts model
-
F.Y. Wu, The Potts model, Rev. Modern Phys. 54 (1982) 235-268.
-
(1982)
Rev. Modern Phys.
, vol.54
, pp. 235-268
-
-
Wu, F.Y.1
|