## 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.

1. 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.

1. First week, August 27.
• Part I, Labeled counting
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.
2. Second week, September 3.
• Part I, Labeled counting
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 Sn. -- 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.
3. Third week, September 10.
• Part I, Labeled counting
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.
4. 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.
5. Fifth week, September 24.
• Part I, Labeled counting
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.
6. Sixth week, October 1.
• Part I, Labeled counting
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 En, 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.
7. Seventh week, October 8.
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.
8. 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.
9. Nineth week, October 22.
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.!
10. 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.
11. 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 + ... + xn. -- 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.
12. Twelfth week, November 12.
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.
13. 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(T2(x) - T(x)2).
14. Forteenth week, December 3.
• Part IV, Graphs
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.! :)

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