ALGEBRAIC
COMBINATORICS
c
C D Go dsil
To Gillian
Preface
There are p eople who feel that a combinatorial result should b e given a
purely combinatorial pro of but I am not one of them For me the most
interesting parts of combinatorics have always b een those overlapping other
areas of mathematics This b o ok is an intro duction to some of the interac
tions b etween algebra and combinatorics The rst half is devoted to the
characteristic and matchings p olynomials of a graph and the second to
p olynomial spaces However anyone who lo oks at the table of contents will
realise that many other topics have found their way in and so I expand on
this summary
The characteristic p olynomial of a graph is the characteristic p olyno
matrix The matchings p olynomial of a graph G with mial of its adjacency
n vertices is
b n 2c
X
k n 2k
G k x p
k =0
where p G k is the numb er of k matchings in G i e the numb er of sub
graphs of G formed from k vertex disjoint edges These de nitions suggest
that the characteristic p olynomial is an algebraic ob ject and the matchings
p olynomial a combinatorial one Despite this these two p olynomials are
closely related and therefore they have b een treated together In devel
oping their theory we obtain as a by pro duct a numb er of results ab out
orthogonal p olynomials The numb er of p erfect matchings in the comple
ment of a graph can b e expressed as an integral involving the matchings
by which we p olynomial This motivates the study of moment sequences
mean sequences of combinatorial interest which can b e represented as the
sequence of moments of some measure
To b e brief if not cryptic a p olynomial space is obtained by asso ciat
ing an inner pro duct space of p olynomials to a combinatorial structure
The combinatorial structure might b e the set of all k subsets of a set of v
elements the symmetric group on n letters or if the reader will b e gener
n
ous the unit sphere in R Given this set up it is p ossible to derive b ounds
of co des and designs in the structure The derivations are on the sizes
very simple and apply to a wide range of structures The resulting b ounds
are often classical the simplest and b est known is Fisher s inequality from
design theory
iv Preface
Polynomial spaces are p erhaps imp ossibly general We distinguish one
imp ortant family which corresp onds when the underlying set is nite to Q
p olynomial asso ciation schemes The latter have a well develop ed theory
thanks chie y to work of Delsarte Our approach enables us to rederive and
extend much of this work In summary the theory of p olynomial spaces
provides an axiomatisation of many of the applications of linear algebra
to combinatorics along with a natural way of extending the theory of Q
p olynomial asso ciation schemes to the case where the underlying set is
in nite
From this discussion it is clear that to make sense of p olynomial spaces
some feeling for asso ciation schemes is required Hence I have included
a reasonably thorough intro duction to this topic To motivate this in
turn I have also included chapters on strongly regular and distance regular
graphs Orthogonal p olynomials arise naturally in connection with p olyno
mial spaces and distance regular graphs and thus form a connecting link
b etween the two parts of this b o ok
My aim has b een to write a b o ok which would b e accessible to b egin
ning graduate students I b elieve it could serve as a text for a numb er of
t courses in combinatorics at this level and I also hop e that it will di eren
prove interesting to browse in The prerequisit es for successful digestion of
the material o ered are
Linear algebra Familiarity with the basics is taken for granted The
sp ectral decomp osition of a Hermitian matrix is used more than once
The theory is presented in Chapter Positive semi de nite matrices
app ear A brief summary of the relevant material is included in the
app endix
Combinatorics The basic language of graph theory is used without
preamble e g spanning trees bipartite graphs and chromatic num
b er Once again some of this is included in the app endix Generating
functions and formal p ower series are used extensively in the rst half
of the b o ok and so there is a chapter devoted to them
Group theory The symmetric group creeps in o ccasionally along
with automorphism groups of graphs The orthogonal group is men
tioned by name at least once
Ignorance By which I mean the ability to ignore the o dd paragraph
devoted to unfamiliar material in the trust that it will all b e ne at
the end
I have not b een able to draw up a dep endence diagram for the chapters
which would not b e misleading This is b ecause there are few chains of
argument extending across chapter b oundaries but many cases where the
Preface v
material in one chapter motivates another For example it should b e
p ossible to get through the chapter on asso ciation schemes without reading
the preceding chapter on distance regular graphs However these graphs
provide one of the most imp ortant classes of asso ciation schemes
By way of comp ensation for the lack of this traditional diagram I
include some suggestions for p ossible courses
The matchings p olynomial and moment sequences
The characteristic p olynomial
Strongly regular graphs distance regular graphs and asso ci
ation schemes
Equitable partitions and co des in distance regular graphs
Polynomial spaces
In making these suggestions I have made no serious attempt to consider
the time it would take to cover the material indicated On the basis of my
own exp erience I think it would b e p ossible to cover at most three pages
p er hour of lectures On the other hand it would b e easy enough to pare
down the suggestions just made For example Chapter covers formal
p ower series and generating functions and dep ending on the backgrounds
of one s victims this might not b e essential in and
Bender Andries I have b een help ed by advice and comments from Ed
Brouwer Dom de Caen Michael Do ob Mark Ellingham Tony Gardiner
y Gillian Nonay Jack Ko olen Gordon Royle Bill Martin Brendan McKa
J J Seidel and Akos Seress Dom in particular has made heroic e orts to
protect me from my own stupidity I am very grateful for all this assistance
I would also like to thank John Kimmel and Jim Geronimo of Chapman and Hall for their part in the pro duction of this b o ok
Contents
Preface vii
Contents xi
The Matchings Polynomial
Recurrences
Integrals
Ro ok Polynomials
The Hit Polynomial
Stirling and Euler Numb ers
Hit Polynomials and Integrals
Exercises
Notes and References
The Characteristic Polynomial
Co e cients and Recurrences
Walks and the Characteristic Polynomial
Eigenvectors
Regular Graphs
The Sp ectral Decomp osition
Some Further Matrix Theory
Exercises
Notes and References
Formal Power Series and Generating Functions
Formal Power Series
Limits
Op erations on Power Series
Exp and Log
Non linear Equations
Applications and Examples
Exercises
Notes and References
viii Contents
Walk Generating Functions
Jacobi s Theorem
Walks and Paths
A Decomp osition Formula
The Christo el Darb oux Identity
ertex Reconstruction V
Cosp ectral Graphs
Random Walks on Graphs
Exercises
Notes and References
of Graphs Quotients
Equitable Partitions
Eigenvalues and Eigenvectors
Walk Regular Graphs
Generalised Interlacing
Covers
The Sp ectral Radius of a Tree
Exercises
Notes and References
Matchings and Walks
The Path Tree
Tree Like Walks
Consequences of Reality
Christo el Darb oux Identities
Exercises
Notes and References
Pfa ans
The Pfa an of a Skew Symmetric Matrix
Pfa ans and Determinants
Row Expansions
Oriented Graphs
Orientations
The Di culty of Counting Perfect Matchings
Exercises
Notes and References
Contents ix
Orthogonal Polynomials
The De nitions
The Three Term Recurrence
The Christo el Darb oux Formula
Discrete Orthogonality
Sturm Sequences
Some Examples
Exercises
Notes and References
Moment Sequences
Moments and Walks
Moment Generating Functions
Hermite and Laguerre Polynomials
The Chebyshev Polynomials
The Charlier Polynomials
She er Sequences of Polynomials
Characterising Polynomials of Meixner Typ e
The Polynomials of Meixner Typ e
Exercises
Notes and References
Strongly Regular Graphs
Basic Theory
Conference Graphs
Designs
Orthogonal Arrays
Exercises
Notes and References
Distance Regula r Graphs
Some Families
Distance Matrices
Parameters
ts Quotien
Imprimitive Distance Regular Graphs
Co des
Completely Regular Subsets
Examples
Exercises
Notes and References
x Contents
Asso ciation Schemes
Generously Transitive Permutation Groups
s p s and q
P and Q Polynomial Asso ciation Schemes
Pro ducts
Primitivity and Imprimitivity
Co des and Antico des
Equitable Partitions of Matrices
Characters of Ab elian Groups
Cayley Graphs
Translation Schemes and Duality
Exercises
Notes and References
Representations of Distance Regular Graphs
Representations of Graphs
The Sequence of Cosines
Injectivity
Eigenvalue Multipliciti es
Bounding the Diameter
Spherical Designs
Bounds for Cliques
Feasible Automorphisms
Exercises
Notes and References
Polynomial Spaces
Functions
The Axioms
Examples
The Degree of a Subset
Designs
The Johnson Scheme
The Hamming Scheme
Co ding Theory
Group Invariant Designs
Weighted Designs
Exercises
Notes and References
Contents xi
Q Polynomial Spaces
Orthogonal Polynomials Zonal
Zonal Orthogonal Polynomials Examples
The Addition Rule
Spherical Polynomial Spaces
Harmonic Polynomials
Asso ciation Schemes
Q Polynomial Asso ciation Schemes
Incidence Matrices for Subsets
v k is Q Polynomial J
Exercises
Notes and References
Tight Designs
Tight Bounds
Examples and Non Examples
The Grassman Space
Linear Programming
Bigger Bounds
Examples
Exercises
Notes and References
Terminology App endix
Index of Symb ols
Index