Topics in Combinatorics, Graph Enumerations
Math. 649, Sec. 001, Fall 2018
Announcements and Notes
Urgent stuff
The last update on the lecture notes!
has been posted here below.!
Dear Folks!
Your over all letter grade has been posted
on the Patriot Web. -- Thank you all for your
dedication and work, it was a pleasure to have
you in my class :) -- Happy Holidays to you all.!
Solutions
to all four howeworks have been poste here below.
General Info
Here below is a pdf version of the promotional
poster for the course, it contains a rough outline
of our plan of attack:
NB!
All handouts for this class will be listed with an itemized icon
as with the Syllabus.
For general detailed information regarding this course, see
the Syllbus here below.
Please read it carefully as it
outlines the overall plan for this course.
It should answer all practical questions.
In the above figure we see all the simple graphs on one, two, three and four
unlabeled vertics; no two graphs are isomophic.
This topics course will serve as an introduction
to the art of counting various types of graph, be it
labelled or unlabeled. This can be very useful when
considering many practical questions, like "Does
there exist a graph with this and that property?"
or "Exactly how many graphs are there satifying this
and that?", "How can we encode the combinatorial results
in a compact form using analytical functions?" . -- I
will assume basic knowledge in algebra
(Math 621, Algebra, first year graduate course, or equivalent),
and some combinatorics (Math 325, Discrete Mathematics II, or equivalent)
is preferred. However, the most important thing is to have mathematical
maturity and an open mind.
There is no required text for the course; the lectures will be self contained.
Standard results that will be stated without proofs will for the most
part be contained in the following book. It is recommended that
you will have easy access to it for reference.
- Thomas W. Hungerford,
Algebra,
(GTM -- 73), Springer Verlag, New York, (2003) or most recent printing.
Weekly recap of Lectures and Problems
Here below is a draft of the lecture notes, together
with the figures, that I have so far. These will be updated
as we go along. -- Please let me know of any typos, snafus etc.!
Here below are some notes to recap the lecture each week.
Also, occasional problems and/or their solutions might
be posted here as well.
- First week, August 27.
We have so far reviewed the definitions of a (general) graph,
a simple graph, and what we mean by a homomorphism from one
graph to another (simple or not) a particular interesting homomorphism
is an isomorphism from one graph to another. When the graphs are simple
we can omit the use of the edge map, and hence the condition for
the ordered tuple to consitute a homomorphism is much simpler.
An isomorphism from a graph to itself is an automorphism.
The problem of checking whether or not a given ordered tuple of vertex
and edge maps constitutes an isomorphism is easy and efficiently checked.
However, the problem of determining whether or not two given graphs
are isomorphic or not (i.e. whether there is an isomorphism from one
to another) is computationally very hard, even harder than exponential
time. -- We will continue our discussion on basic properties before
venture into explicit counting.
- Second week, September 3.
Isomorphism relation between graphs in any collection/set of
graphs is an equivalence relation, and hence yields a partition
of the given set of graphs, in such a way that each part
contains all the graphs that are isomorphic. For a fixed graph
G the set of its automorphism Aut(G) forms
a group. Further, for a fixed vertex
labeling of its n vertices, Aut(G) can always
be viewed as a subgroup of the symmetric group
S_{n}. -- Many counting problems we encounter in
graph theory, involve counting a set S(G) associated to
a given fixed graph G. For example, the number of spanning
trees, the number of vertex colorings using a specific amount of colors,
the number perfect matchings of the graph etc. What we like to do, however,
is to concentrate on counting the number of graphs on a bounded number
of vertices (and/or edges) that have a certain property. In a
labeled counting two graphs on a fixed set of vertices are considered the same
iff for each pair of vertices (distinct or not) the same number of
edges connect the pair in both graphs.
- Third week, September 10.
We presented some explicit formulae for the number of graphs on a fixed
number of labeled vertices, simple or not. In an unlabeled counting
two graphs on a fixed set of vertice are considered the same iff they
are isomorphic. -- A good tool for enumeration, labeled or unlabeled,
is that of generating functions. For now, we will be dealing mostly
with two kinds: (i) the usual generating function for a number sequence,
and (ii) the exponential generating function for a number sequence. Regardless
of usual convergence of these functions as series, we can always manipulate
them formally without any worry. We discussed a few examples of how
generating functions can be used for encoding and also for finding
explicit formulae for discrete number sequences that are presented
recursively, in particular that of the Catalan numbers. Note, that even
if we knew beforehand the simple formula for the n-th
Catalan number, it is quite difficult to prove it using induction.
- Fourth week, September 17.
When dealing with generating functions, it is important to note
that we are solely doing so formaly, and we are not concerned by
their convergence as real or complex power series functions. This is quite
OK, as long as in our formal arithmetic, all our coefficients are the result
of finitely many ringtheoretic operations. By a weak ordered partition
of a finite set into k parts, we mean an ordered k-tuple
of disjoint subsets of our finite set, whose union is the whole set. Here some
of the sets can be empty. In contrast a proper partition is an unordered
collection of disjoint proper subsets whose union is the whole set. The
Stirling numbers of the 2nd kind S(n,k) tally the number
of proper partitions of an n-element set into k proper parts.
They satisfy a nice binomial-like recursion. The Bell numbers B(n)
is the number of unrestricted proper partitions of an n-element set
into parts. -- The Labeled Counting Lemma (LCL) is fundamental
in labeled graph counting. It describes the combinatorial meaning
of the coefficients of the product of two exponential generating functions
of two discrete number sequences. Next week we will look at some examples and
some generatlizations.
- Fifth week, September 24.
We have seen some examples of how the LCL can yield a nice
closed formula for the exponentila generating function for
a counting sequence that has no nice formula for its n-th
term. Similarly the Generalized Labeled Counting Lemma (GLCL)
can give a closed formula for the exp. gen. function for a a counting
sequence that also has no nice formula for its n-th term. What
this means is that there is a fairly efficient procedure to compute each
fixed term, despite the lack of closed formula. Note! Both for the LCL
and the GLCL it is of essence that the graph theoretic properties are
mutually exclusive! When considering the same graph theoretic property
for unrestricted partitions of a set of vertices, we can use
The Exponential Formula (ExpF) to obtain the exp. gen. function
for a counting sequence, if we know the number for each part of
the partition. Moreover, we can use the ExpF to
solve it for the exp. gen. function in the exponent,
in terms of the exp. gen. function on the right. In this way we
obtain the exp. gen. function for the number of connected simple
graphs on n labeled vertices (Riddel's theorem since 1951).
Further, by taking the formal derivative, we can obtain a nice
recursion of the exponent counting sequence using the ExpF.
- Sixth week, October 1.
There is a nice bijection between the collection of simple
graphs on n-1 vertices and simple even graphs
on n vertices. Since a graph is Eulerian
if and only if it is even and connected, we were able
to use the ExpF to find the exponential generating function
for all simple Eulerian graph on n labeled vertices,
and further we were able to find a nice recursion for
E_{n}, the number of Eulerian
graphs on n labeled vertices.
- Part II, Unlabeled counting
To count unlabeled graphs, we reformulate the problem we have
to the problem of finding the number of orbits of the appropriate
permutation group. Then we will use Burnside Lemma to
express the number of orbits in terms of the number of objects
that are fixed by the permutation of our group. -- We recall
that two groups are isomorphic if there is a bijective group
homomorphism from one to the other. Further, two isomorphic
permutation groups are identical if they act on
sets of the same cardinality in such a way the group action respects
the cardinality/labelning mapping from one set to the other. This
is, of course, a stronger condition, and works only if we are
dealing with permutation groups, but not abstract groups.
For two isomorphic simple graphs, their automorphism groups
and the automorphism groups of their complements are all
indeed identical permutation groups. -- The cycle index
of a permutation group is a polynomial in n
variables with its number of terms equal to the order of
the group. We will work with this index all next week,
and then move onto Burnside Lemma.
- Seventh week, October 8.
- Part II, Unlabeled counting
The cycle index of a permutation group is quite messy,
but we will see that it does have some nice combinatorial
interpretations. We have so far computed the cycle index
for some common permutation groups, like the symmetric
group of degree n, its alternating group,
the cycle group of order n, and the dihedral
group of degree n. Next we will discuss Burnside
Lemma and see how the cycle index is usefull.
- Eighth week, October 15.
- Part II, Unlabeled counting
Despite the intricate structure of the cycle index of a
permutation group, it does not determine the group; there
are non-isomorphic groups, acting on sets of the same cardinality,
that have the same cycle index! -- The Burnside Lemma (BL)
was known to Frobenius and others, and hence is sometimes called
"the lemma that is not Burnside's", is a result that relates the
number of orbits of a group acting on a set with the number of
1-cycles of the group elements, i.e. the number of elements of
the set that are left fixed by group elements. This lemma has
both a restricted version and a more general weighted version,
both of which will come in handy when discussing Polya's Theorem.
- Nineth week, October 22.
- Part II, Unlabeled counting
The main theorem of this course is certainly
Polya's Enumeration Theorem (PET). But it is, however,
not a deep theorem; the main ingredient (that is, the deepest
mathematical theorem it relies on) is Burnside Lemma (BL) and
the rest is more interpretation what the coefficients mean
when we apply the cycle index to the figure counting series
b(x). As a first example we used PET to count
the number of "necked" necklaces (not allowed to take the necklace
of the person wearing it; i.e. reflective images are
considered distinct) where the beads have two colors, black
and white. We obtained an explicit formula for the number
of necked necklaces on n beads, where k
of them are black. -- We will continue to use PET for
various counting purposes, more or less for the rest of
this semester.!
- Tenth week, October 29.
- Part II, Unlabeled counting
We have seen some examples on how PET can be used to count
various finite combinatorial situations, where we view two
of them to be the same if for a given permuatation group
they differ by a permuation. To tally all of the possibilities
we simply evaluate C(1), that is, we put the cardinality
of the target set Y into the cycle index. -- PET
has a multivariable generalization, and its proof is similar
to that of PET; the main mathematical ingredient is
(as before with PET) BL, but it is just messier. However,
the Multivariable PET (MVPET) is used when we color
our items (vertices, beads, etc.) with more than two colors.
The algebra will be quite more involved, but it works the same
way as the one variable version. -- Next week we will discuss
the in more detail the special configuration series (or polynomial)
1+x which plays a special role when counting.
- Eleventh week, November 5.
- Part II, Unlabeled counting
The configuration series b(x) = 1+x plays a special role
when counting, since it represents the "boolean" weight;
into the set containing 0 and 1,
or "true" and "false", etc. It can be used to count k-element
subsets that are equivalent w.r.t. a permutation group A.
If n is at least 3, and we have two k-element
subsets S and S' in the set of 1,...,n,
there is always an even permutation that maps S to S'.
This means that the cycle index of both the symmetric group and the
alternating group of degree n evaluated at b(x) = 1+x
is particularly nice as they both evaluate to
1 + x + ... + x^{n}. -- Counting injective functions
from X to Y comes in handy when we want to count
n-element subsets of Y. For example, if we know
the number of simple, connected unlabeled graphs on a given number of
vertices, we can use this technique to count the number of simple
unlabeled graphs that have exactly three, say (or any given fixed number),
distinctly shaped components. Such enumeration can become cumbersome,
so it is important to keep track of what we know, so we can use
that elsewhere when needed.
- Twelfth week, November 12.
- Part III, Trees and Forests
We just started discussing enumeration of unlabeled trees on
a given number of vertices. As so often, when dealing with
enumerative problems involving trees, it is best to consider
rooted trees on a given number of unlabeled vertices first.
This we did and instead of obtaining a direct formula for
the function counting series T(x) for rooted
unlabeled trees, we obtained the "next best thing", namely
an exponential equation for T(x). This we will use
to (i) obtain recursive equations for the coefficients of
T(x) (so we can compute the number of rooted trees on
a given number of unlabled vertices recursively), and (ii)
to obtain a formula for number of (unrooted) trees on
a given number of unlabeled vertices.
- Thirteenth week, November 19.
- Part III, Trees and Forests
We have seen that the equation for T(x) can be
used to obtain a recursion for its coefficents, namely
the number of rooted unlabeled trees. Although this recursion
looks involved at first sight, it is relatively fast to
compute each given term from the previous ones. We also
derived an equivalent equation for T(x) due
to Cayley from 1897. -- To derive a formula for the number
of unrooted unlabeled trees, we first had to establish
a lemma on the number of dissimilar vertices in terms
of dissimilar vertices in each block-class. It then
turns out that if t(x) is the generating
function for the number of unrooted unlabeled trees,
then an elegant formula due to Richart Otter (1948)
states that
t(x) = T(x) - 1/2(T^{2}(x) - T(x)^{2}).
- Forteenth week, December 3.
Having derived a nice formula (or at least an equation
for getting the formula) for both rooted unlabeled trees
and then unrooted unlabeled trees, we delve into the
last topic of this course, namely to enumerate graphs,
first simple graphs and then general graphs. When
enumerating simple graphs it is necessary to consider
the pair group of a permutation group, which
is just the permutation group together with its
action on the set of two-element subsets of the original
set X. Since there is a 1-1 correspondence
between simple graphs and functions from the two-element
subsets of X to the two element set containing zero and one,
then we can enumerate the simple
graphs in terms of the cycle index of this corrssponding
pair group. The theorem (originally by Redfield from 1927)
shows how involved this cycle index really is, but
from a computational point of view it is not much worse
than the cycle index of the symmetric group. We
demonstrated this proof by an example.
That is it folks! I hope you have enjoyed the class and
gained some from it during the semester. In particular, I
hope I have conveyed the idea that when counting graphs
in general, one cannot always expect simple nice formulas;
there simply aren't enough functions in our arsenal to
be able to do just that. However, by reducing the overall
complexity to some "known" complex entities (like cycle indices
of known groups and pair groups) we can express the number of
this and that in terms of those. -- You have all been true
troopers to come every week to this topics class
and actively participate. Your grades will be posted on
the GMU Patriot web. --
Happy Holidays!
If you have questions or want to chat, please drop me and email.! :)
EOF/EOS