Mathematical Publications of  Norman L Biggs

Books Papers Miscellaneous Publications Contributions to the History of Mathematics Book Reviews

Books on Mathematics

Finite Groups of Automorphisms. L.M.S. Lecture Notes Series, No. 6. Cambridge University Press, 1971.

Algebraic Graph Theory. Cambridge Tracts in Mathematics No. 67, Cambridge University Press, 1974.

Graph Theory 1736-1936 (with E.K. Lloyd and R.J. Wilson). Oxford University Press, 1976. Second edition 1986, Japanese edition 1986.

Interaction Models. L.M.S. Lecture Notes Series, No. 30. Cambridge University Press, 1977.

Permutation Groups and Combinatorial Structures (with A.T. White). Cambridge University Press, 1979. Chinese edition 1988.

Discrete Mathematics. Oxford University Press, 1985, revised edition 1989, reprinted many times. Spanish edition 1994.

Introduction to Computing with Pascal. Oxford University Press, 1989.

Computational Learning Theory: an Introduction (with M. Anthony). Cambridge Tracts in Theoretical Computer Science, No. 30. Cambridge University Press, 1992. Revised edition 1997.

Algebraic Graph Theory (Second Edition). Cambridge Mathematical Library. Cambridge University Press, 1993.

Mathematics for Economics and Finance (with M. Anthony). Cambridge University Press, 1996. Chinese edition 1998. Japanese edition 2000.

Discrete Mathematics (Second Edition). Oxford University Press, 2002. Website.

Codes: An Introduction to Information Communication and Cryptography . Springer 2008. Website.

Quite Right: The Story of Mathematics, Measurement, and Money. Oxford University Press, 2016.

Papers on Mathematics

Year Published: 1969 1971 - 79  |  1980 - 89  |  1990 - 1999  |  2000 - 


Rectangulations. Proc. Cambridge Philos. Soc. 65 (1969) 399-408.


Intersection matrices for linear graphs. In: Combinatorial Mathematics and its Applications (ed. D.J.A. Welsh), Academic Press (1971) 15-23.

Spanning trees of dual graphs. J. Combinatorial Theory (B), 11 (1971) 127-131.

Automorphisms of imbedded graphs. J. Combinatorial Theory (B), 11 (1971) 132-138.

On trivalent graphs (with D.H. Smith). Bull. London Math. Soc., 3 (1971) 155-158.

Classification of complete maps on orientable surfaces. Rendiconti di Matematica (Rome), 4 (1971) 645-655.


Recursive families of graphs (with R.M. Damerell and D.A. Sands). J. Combinatorial Theory (B), 12 (1972) 123-131.

Cayley maps and symmetrical maps. Proc. Cambridge Philos. Soc., 72 (1972) 381-386.

On the symplectic representation of map automorphisms. Bull. London Math. Soc., 4 (1972) 303-306.

An edge-colouring problem. Amer. Math. Monthly, 79 (1972) 1018-1020.

Pictures. In: Combinatorics, (ed. D.J.A. Welsh and D.R. Woodall), Institute of Mathematics and its Applications (1972) 1-17.


Three remarkable graphs. Canadian J. Math., 25 (1973) 397-411.

Expansions of the chromatic polynomial. Discrete Mathematics, 6 (1973) 105-113.

Perfect codes in graphs. J. Combinatorial Theory (B),15 (1973) 289-296.


On the symmetry of line graphs. Utilitas Math., 5 (1974) 113-121.

Perfect codes and distance-transitive graphs. In: Combinatorics, (ed. V.C. Mavron and T. McDonough), L.M.S. Lecture Note Series, No. 13 (1974) 1-8.


Designs, factors and codes in graphs. Quart. J. Math. (Oxford), 26 (1975) 113-119.

A theorem on planar partitions (with G.H.J. Meredith). Proc. 5th British Combinatorial Conference, (1975) 73-78.

Approximations for chromatic polynomials (with G.H.J. Meredith). J. Combinatorial Theory (B), 20 (1976) 5-19.


Automorphic graphs and the Krein condition. Geometriae Dedicata, 5 (1976) 117-127.

On the duality of interaction models. Math. Proc. Cambridge Philos. Soc., 80 (1976) 429-436.


Colouring square lattice graphs. Bull. London Math. Soc., 9 (1977) 54-56.


Cluster expansions in graph theory and physics. Quart. J. Math. (Oxford), 29 (1978) 159-174.


On the algebra of graph types. In: Graph Theory and Related Topics (ed. J.A. Bondy and U.S.R. Murty), Academic Press (1979) 81-89.

Some odd graph theory. Annals of the New York Academy of Sciences, 319 (1979) 71-81.

Resonance and reconstruction. In: Surveys on Combinatorics (ed. B. Bollobas), Cambridge University Press (1979) 1-21.


Girth, valency and excess. Linear Algebra and its Applications, 31 (1980) 55-59.

Graphs with even girth and small excess (with T. Ito). Math. Proc. Cambridge Philos. Soc., 88 (1980) 1-10.

A trivalent graph with girth 9 and 58 vertices (with M. Hoare). Discrete Math., 30 (1980) 299-301.


Aspects of symmetry in graphs. In: Algebraic Methods in Graph Theory (ed. L. Lovász and V. Sos), North-Holland (1981) 27-35.

Covering biplanes. In: Theory and Applications of Graphs (ed. G. Chartrand et al.), Wiley (1981) 73-82.

Covering graphs and symmetric designs (with T. Ito). In: Finite Geometries and Designs, (ed. P.J. Cameron et al.), Cambridge University Press (1981) 40-51.


Excess in vertex-transitive graphs. Bull. London Math. Soc., 14 (1982) 52-54.

Distance-regular graphs with diameter 3. Ann. Discrete Math., 15 (1982) 69-80.

Constructing 5-arc-transitive cubic graphs. J. London Math. Soc. (2), 26 (1982) 193-200.

A new 5-arc-transitive cubic graph. J. Graph Theory, 6 (1982) 447-451.


The sextet construction for cubic graphs (with M.J. Hoare). Combinatorica, 3 (1983) 153-165.


Presentations for cubic graphs. In: Computational Group Theory (ed. M.D. Atkinson). Academic Press (1984) 57-63.

Homological coverings of graphs. J. London Math. Soc. (2), 30 (1984) 1-14.

Rotations and graphs of large girth (with J. Shawe-Taylor). In: Geometrical Combinatorics (ed. F.C. Holroyd and R.J. Wilson), Pitman (1984) 1-9.


Infinite covers of cages (with S.K. Burford). Europ. J. Combinatorics, 6 (1985) 7-12.


Cubic distance-regular graphs (with A.G. Boshier and J. Shawe-Taylor) J. London Math. Soc. (2), 33 (1986) 385-394.


The spectral radius of infinite graphs (with B. Mohar and J. Shawe-Taylor) Bull. London Math. Soc., 20 (1988) 116-120.

Girth and residual finiteness. Combinatorica, 8 (1988) 307-312.

Graphs with large girth. Ars Combinatoria, 25-C (1988) 73-80.


Cubic graphs with large girth. Annals of the New York Academy of Sciences, 555 (1989) 56-62.

Confluence of some presentations associated with graphs. Discrete Math., 75 (1989) 56-62.

A proof of Serre's theorem. Discrete Math., 78 (1989) 55-57.

The growth rate of the harmonious chromatic number (with D. Beane and B.J. Wilson). J. Graph Theory, 13 (1989) 291-299.


Note on the girth of Ramanujan graphs (with A.G. Boshier). J. Combinatorial Theory (B), 49 (1990) 190-194.

Some heuristics for graph colouring. In: Graph Colourings (ed. R. Nelson and R.J. Wilson), Pitman Research Notes in Mathematics 218, Longmans (1990) 87-96.

The learnability of formal concepts (with M.H.G. Anthony and J. Shawe-Taylor). Proceedings of the Third Annual Workshop on Computational Learning Theory (COLT 1990), M.A. Fulk (ed.), Morgan-Kaufmann, San Mateo Ca. (1990) 246-257.


Learning algorithms: theory and practice. In: Neural Network Applications (ed. J.G. Taylor), Springer (1992) 1-11.

Theoretical and practical studies of a competitive learning process (with G.R. Brightwell and D. Tsoubelis). Network 3 (1992) 285-301.

A framework for cumulative learning. Mathematics Preprint Series, LSE-MPS-27, London School of Economics, 1992.


Bounding sample size with the Vapnik-Chervonenkis dimension (with M.H.G. Anthony and J. Shawe-Taylor). Discrete Applied Mathematics 42 (1993) 65-73.

The mean chromatic number of paths and cycles (with M.H.G. Anthony). Discrete Mathematics 120 (1993) 227-231.

Computational learning theory for artificial neural networks (with M.H.G. Anthony). In: Mathematical Approaches to Neural Networks (ed. J.G. Taylor), North-Holland (1993) 25-62.

Discrete Mathematics in Finance: Two Applications, Mathematics Preprint Series LSE-MPS- 49, London School of Economics, 1993.

Potential theory on distance-regular graphs. Combinatorics, Probability and Computing 2 (1993) 243-255.


Combinatorics and connectionism. Discrete Math. 124 (1994) 19-36.

Exchange rate networks and mechanisms. Mathematics Preprint Series, LSE-MPS-64, London School of Economics, December 1993, revised May 1994.

PAC learning and artificial neural networks (with M.H.G. Anthony). In: The Handbook of Brain Theory and Neural Networks, (ed. M.A. Arbib), Bradford/MIT Press (1994).

Exchange rates and the matrix-tree theorem. Theoretical Economics Discussion Papers TE/94/273, London School of Economics, 1994.

How to compute the spectral density of a lattice and its quotients. Mathematics Preprint Series, LSE-MPS-74, London School of Economics, October 1994.


The spectral density of covering lattices. Mathematics Preprint Series, LSE-MPS-78, London School of Economics, January 1995.

A computational learning theory view of economic forecasting with neural nets (with M.H.G. Anthony). In: Neural Networks in the Capital Markets (ed.A.Refenes), Wiley (1995) 77-98.

Algorithms for learning and coloring. In: Graph Theory, Combinatorics and Applications, (ed. Y. Alavi and A. Schwenk), Wiley (1995) 63-70.

The potential of potential theory. LSE Mathematics Preprint Series, LSE-MPS-91, September 1995.


Chip-firing on distance-regular graphs. CDAM Research Report Series, LSE-CDAM-96-11, June 1996.


International Finance. In: Graph Connections (ed. L.W. Beineke and R.J. Wilson), Oxford University Press (1997) 261-279.

Chip-firing and the chromatic polynomial (with P. Winkler). CDAM Research Report Series, LSE-CDAM-97-03, February 1997.

Integer programming techniques for the frequency assignment problem. CDAM Research Report Series, LSE-CDAM-97-06, April 1997.

Algebraic potential theory on graphs. Bull. London Math. Soc. 29 (1997) 641-682


Constructions for cubic graphs with large girth. Electronic Journal of Combinatorics 5 (1998) A1. PDF.

Optimising the signal-to-noise ratio (with D Fon der Flaass). CDAM Research Report Series, LSE-CDAM-98-16, August 1998.


Chip-firing and the critical group of a graph. J. Algebraic Combinatorics, 9 (1999) 25-45.

The Tutte polynomial as a growth function. Journal of Algebraic Combinatorics 10 (1999) 115-133.

The chromatic polynomial of the 3 x n toroidal square lattice. CDAM Research Report Series, LSE-CDAM 99-05, June 1999.

T=0 partition functions for Potts antiferromagnets on square lattice strips with (twisted) periodic boundary conditions (with R.Shrock). J.Phys. A. 32 (1999) L489-L493.


A matrix method for chromatic polynomials – II. CDAM Research Report Series, LSE-CDAM 2000-04, April 2000.

(with P.Reinfeld). The chromatic roots of generalised dodecahedra. CDAM Research Report Series, LSE-CDAM 2000-07, June 2000.


Equimodular curves for reducible matrices CDAM Research Report Series, LSE-CDAM 2001-01, January 2001. PDF.

A matrix method for chromatic polynomials. J. Combinatorial Theory (B), 82 (2001) 19-29.


Chromatic polynomials for twisted bracelets. Bull. London Math. Soc. 34 (2002) 129-139.

Chromatic polynomials and representations of the symmetric group. Linear Algebra and its Applications 356 (2002) 3-26.

Equimodular curves. Discrete Mathematics 259 (2002) 37-57.


Algebraic methods for chromatic polynomials (with M H Klin and P Reinfeld). Europ. J. Combinatorics 25 (2004) 147-160.

Specht modules and chromatic polynomials. J. Combinatorial Theory (B) 92 (2004) 359 - 377.


Chromatic polynomials of some families of graphs I: Theorems and Conjectures. CDAM Research Report Series, LSE-CDAM 2005-09, May 2005. PDF.


The critical group from a cryptographic perspective. Bull. London Math. Soc. , 39 (2007) 829-836.


Chromatic Roots of the Quartic Mobius Ladders CDAM Research Report LSE-CDAM 2008-05, May 2008. PDF.

A Matrix Method for Flow Polynomials. CDAM Research Report LSE-CDAM 2008-08, June 2008. PDF.


Tutte Polynomials of Bracelets. CDAM Research Report LSE-CDAM-2009-01, January 2009. PDF.

Strongly Regular Graphs with No Triangles. Research Report, September 2009. arXiv:0911.2160v1 PDF.

Families of Parameters for SRNT Graphs. Research Report, October 2009. arXiv:0911.2455v1 PDF.


Tutte Polynomials of Bracelets. J. Algebraic Combinatorics 32 (2010) 389-398.

The Second Subconstituent of some Strongly Regular Graphs. Research Report, February 2010. arXiv:1003.0175v1 PDF.


Some Properties of Strongly Regular Graphs. Research Report, May 2011. arXiv:1106.0889v1 PDF.


Applications of Integer Programming Methods to Cages (with F. de Ruiter). Electronic J. Combinatorics 22(4) (2015) #P4.35. PDF.


Chromatic polynomials and toroidal graphs. Australasian Journal of Mathematics 67(2) (2017) 235-242.


Miscellaneous Publications

Directional guidance of motor vehicles. Ergonomics, 9 (1966) 193-202.

Undergraduate projects in mathematics (with K.E. Hirst) Educational Studies in Mathematics 1 (1969) 252-261.

Three notable points in a triangle. Mathematical Gazette 54 (1970) 63-64.

Combinatorics: a guide to the literature (with R.P. Jones). In: The Use of Mathematical Literature (ed. A.R. Dorling), Butterworths (1977).

The schoolgirls problem. Math. Spectrum, 12 (1979) 7-11.

Derek Arthur Waller [Obituary]. Bull. London Math. Soc., 12 (1980) 308-309.

British Combinatorics in Ancient Times (1969-77). British Combinatorial Bulletin 1997, 10-13.

Professor W.T. Tutte [Obituary]. The Independent, 9 May 2002.

W.T. Tutte 1917-2002. In: Surveys in Combinatorics, (Cambridge University Press), 2003, 3-9.

Tutte, William Thomas. In: Oxford Dictionary of National Biography Oxford U.P, 2008.

Tutte, William (Bill) Thomas. In: New Dictionary of Scientific Biography, Scribner 2008.

Contributions to the History of Mathematics

C.S. Peirce and De Morgan on the four-colour conjecture (with E.K. Lloyd and R.J. Wilson) Historia Math., 4 (1977) 215-216.

The roots of combinatorics. Historia Math., 6 (1979) 109-136.

T.P. Kirkman, mathematician. Bull. London Math. Soc., 13 (1981) 97-120.

De Morgan on map colouring and the separation axiom. Archive for Hist. Exact Sci., 28 (1983) 165-170.

The historical background to Gerhard Ringel's work (with E.K. Lloyd and R.J. Wilson). In: Topics in Combinatorics and Graph Theory (ed. R. Bodendiek and R.Henn) Physica-Verlag, Heidelberg (1990) 93-99.

Möbius and the Development of Topology in the Nineteenth Century. In: Möbius and his Band (ed. J. Fauvel, R. Flood and R.J. Wilson), Oxford U.P. 1993, 104-119.

The history of combinatorics (with E.K. Lloyd and R.J. Wilson). In: Handbook of Combinatorics (ed. R.L. Graham, M. Grötschel and L. Lovász), North-Holland, 1995.

The Icosian Calculus of Today. Proceedings of the Royal Irish Academy 95A (1995) 23-34.

Mathematics of Currency and Exchange: Arithmetic at the end of the Thirteenth Century. BSHM Bulletin: Journal of the British Soc. Hist. Math. 24 (2009) 67-77.

Applicable Mathematics in the 18th Century: an example from the textile trade. Mathematics Today. 47(2) (2011) 96-100. PDF.

Block Designs (with R.J.Wilson). In: Combinatorics - Ancient and Modern (ed. R.J. Wilson and J.J. Watkins) Oxford 2013, pp.231-250.

Thomas Harriot on continuous compounding. BSHM Bulletin: Journal of the British Society for the History of Mathematics 28 (2013) 66-74. PDF.

More 17th-century Networks. BSHM Bulletin: Journal of the British Society for the History of Mathematics 32(1) (2017) 30-39

Book Reviews

Joseph Hammer: 'Unsolved Problems Concerning Lattice Points', The Times Higher Education Supplement, 1977.

J.A. Bondy and U.S.R. Murty: 'Graph Theory with Applications', International Journal of Mathematical Education in Science and Technology,1978.

P.J. Cameron and J.H. Van Lint: 'Graphs, Codes and Designs', Bull. London Math. Soc., 13 (1981) 178.

Jean Dieudonne: 'A Panorama of Pure Mathematics', Bull. London Math. Soc., 15 (1983) 161-163.

Rudolf Lidl and Gunter Pilz: 'Applied Abstract Algebra', Bull. London Math. Soc., 17 (1985) 490.

E.L. Lawler, J.K. Lenstra, A.H.G. Rinnoy Khan and D.B. Shmoys (eds.): 'The Travelling Salesman Problem - A Guided Tour of Combinatorial Optimization', Bull. London Math. Soc., 18 (1986) 514-515.

P.G. Doyle and J.L. Snell: 'Random Walks and Electrical Networks' and R. Honsberger: 'Mathemtical Gems III', Bull. London Math. Soc. 19 (1987) 104-105.

V. Krishnamurthy: 'Combinatorics: Theory and Applications', Bull. London Math. Soc., 19 (1987) 194.

L.Lovasz and M.D. Plummer: ' Matching Theory' Bull. London Math. Soc. 20 (1985) 272-273.

T. Nishizeki and N. Chiba: 'Planar Graphs: Theory and Algorithms', Bull. London Math. Soc., 21 (1989) 497-498.

M. Grötschel, L. Lovász and A Schrijver: 'Geometric Algorithms and Combinatorial Optimization', Bull. London Math. Soc., 22 (1990) 204-205.

A.E. Brouwer, A.M. Cohen and A. Neumaier: 'Distance-Regular Graphs', Bull. London Math. Soc., 23 (1991) 305-307.

E.F. Assmus and J.D. Key: 'Designs and their Codes', Bull. London Math. Soc., 25 (1993) 613-614.

I.A. Faradzev, A.A. Ivanov, M.H.Klin and A.J. Woldar (eds.): 'Investigations in the Algebraic Theory of Combinatorial Objects', Bull. London Math. Soc., 27 (1995) 609-610.

S. Golomb: 'Polyominoes' (revised and enlarged edition), London Mathematical Society Newsletter, No.242, October 1996.

F. Chung: 'Spectral Graph Theory', and D. Cvetkovic, P. Rowlinson, and S. Simic: 'Eigenspaces of Graphs', Bull. London Math. Soc. 30 (1998), 197-199.

C. Colbourn and J.H. Dinitz.: 'The CRC Handbook of Combinatorial Designs', Mathematical Gazette March 1997.

J. Robertson and W. Webb: 'Cake-Cutting Algorithms' and S. Brams and A.D. Taylor: 'Fair Division', London Mathematical Society Newsletter, No. 269, March 1999.

J. Gray: The Hilbert Challenge. London Mathematical Society Newsletter No. 295, July 2001.

J.K. Percus: 'The Mathematics of Genome Analysis', London Mathematical Society Newsletter, No.306, June 2002.

A. Hoffman: 'Selected Papers', London Mathematical Society Newsletter No.323, February 2004.

L.W. Beineke and R.J. Wilson (eds.): 'Topics in Algebraic Graph Theory', Combinatorics Probability and Computing, 16 (2007) 171-172.

J.P.S. Kung, G-C. Rota, C.H. Yan: 'Combinatorics: The Rota Way', Bull. London Math. Soc. 2011; doi:10.1112/blms/bdr016.

A. Benjamin, G. Chartrand,, P. Zhang: 'The Fascinating World of Graph Theory', London Mathematical Society Newsletter No.450, September 2015.

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