**Minicourse 1: Another viewpoint of Euler graphs and Hamiltonian graphs **

**Lecturer:** Lih Hsing Hsu (Providence University, Taiwan)

**Syllabus:** It may appear that there is little left to do in regards
to the study of the Hamiltonian property of vertex transitive graphs
unless there is a major breakthrough on the famous Lovasz conjecture.
However, if we extend the concept of the traditional Hamiltonian property
to other Hamiltonicity properties, then there is still much left to explore.
In this series of lectures, I will introduce some of these Hamiltonicity properties,
namely fault tolerant Hamiltonian, spanning connectivity, and mutually independent
Hamiltonicity.

(Another viewpoint of Euler graphs and Hamiltonian graphs)

Ch 1 Euler Graphs

Ch 2 Hamiltonian cycle

Ch 3 Hamiltonian paths

Ch 4 Fault tolerant Hamiltonian cycles

Ch 5 Globally 3connected

Ch 6 Bipartite variations

Ch 7 Cubic 4-ordered Hamiltonian

Ch 8 Mutually Independent Paths and Mutually Independent cycles

Chapter 9 2-cycle and 2-path spanning

Hamiltonian cycle part

Nov122009

Generalized honeycomb torus

Faithful

Dobleloop

**Minicourse 2: The Cayley Isomorphism Problem **

**Lecturer:** Ted Dobson (Mississippi State University, USA, University of Primorska, Slovenia)

**Syllabus:**
In 1967 Ádám conjectured that two circulant graphs Cay(_{n}, S) and Cay(_{n}, T) are isomorphic
if and only if there exists m ∈ _{n}^{*}
such that mS = T. While this conjecture is not true
(although from two different points of view it is mostly true),
the conjecture was quickly generalized to ask
for which groups G any two Cayley graphs Cay(G, S) and Cay(G, T) are isomorphic if and only if
they are isomorphic by an automorphism of G (or α(S) = T for some automorphism α ∈ Aut(G)).
Such a group G is a **CI-group with respect to graphs**. It is easy to show that α(Cay(G, T)) is
a Cayley graph of G for every subset T of G and α ∈ Aut(G), so in testing isomorphism between
two Cayley graphs of a group G one must always check to see if the automorphisms of G are
isomorphisms. From this point of view, asking whether or not a group is a CI-group with respect
to graphs is the same as asking if the minimal or necessary list of permutations that must be
checked as possible isomorphisms is also a sufficient list of permutations to check. We will develop
some of the main tools that are used to determine if a group is a CI-group with respect to graphs,
along with appropriate permutation group theory. The groups G we will focus on will mainly be
of small order (where small order means that there are not many prime factors). These groups are
rich enough to illustrate some, but not all, of the proof techniques that have been developed to
show a group is a CI-group with respect to graphs as well as to highlight some of the obstacles for a
group to be a CI-group with respect to graphs. We will also discuss how the techniques developed
to attack the Cayley isomorphism problem can be modified to attack the isomorphism problem
from graphs that are highly symmetric but not Cayley graphs nor even vertex-transitive, as well
as to attack similar isomorphism problems for other classes of combinatorial objects.

Minicourse

The Cayley Isomorphism Problem Exercises 1 (pdf)

The Cayley Isomorphism Problem Exercises 2 (pdf)

** Minicourse 3: Y-groups via Majorana Theory **

**Lecturer:** Alexander A. Ivanov (Imperial College London, UK)

**Syllabus:** Motivated by an earlier observation by B. Fischer, around 1980
J. H. Conway conjectured that a specific Coxeter diagram Y_{443}
together with a single additional (so-called "spider") relation
form a presentation for the direct product of the largest sporadic
simple group known as the Monster and a group of order 2.
This conjecture was proved by S. P. Norton and the lecturer in 1990.
It appears promising to revisit this subject through currently
developing axiomatic approach to the Monster and its non-associative
196884-dimensional algebra, which goes under the name "Majorana Theory".

**References:**

A. A. Ivanov, Y-groups via transitive extension, *Journal of Algebra*, **218** (1999) 412-435. (pdf)

A. A. Ivanov, *Geometry of Sporadic Groups I. Petersen and Tilde Geometries*, Publisher: Cambridge University Press, 1999.

A. A. Ivanov and S. V. Shpectorov, *Geometry of Sporadic Groups II. Representations and Amalgams*, Publisher: Cambridge University Press, 2002. (pdf)

A. A. Ivanov and S. V. Shpectorov, Amalgams determined by locally projective actions, *Nagoya Math. J.*, **176** (2004) 19-98. (pdf)

A. A. Ivanov, Constructing the Monster amalgam, *Journal of Algebra*, **300** (2005) 571-589. (pdf)

A. A. Ivanov and A. Seress, Majorana representations of A5, *Mathematische Zeitschrift*, **272** (2012) 269-295. (pdf)

Y-groups via Majorana Theory Problem Exercises 1 (pdf)

Y-groups via Majorana Theory Problem Exercises 2 (pdf)

**Minicourse 4: Graphs and their eigenvalues**

**Lecturer:** Bojan Mohar (Simon Fraser University, Canada)

**Syllabus:** The minicourse consists of the following 8 lectures:

1-2. Adjacency matrix and its eigenvalues (basic properties,
Perron-Frobenius theory, interlacing, quotients and equitable partitions, distance-regular graphs). Algebraic and combinatorial properties.

3. Laplacian and expansion (Laplacian matrix, expansion lemma, expanders and Ramanujan graphs).

4-5. Random graphs (random graphs and random matrices, Wigner's semicircle theorem, extensions, quasirandom graphs).

6-7. Applications (Huckel theory, HOMO-LUMO separation, perfect graphs, more on expanders, regularity lemma and graph limits).

8. Hermitian adjacency matrix of a digraph.

Spectral Graph Theory Problem Exercises 1 (pdf)

Spectral Graph Theory Problem Exercises 2 (pdf)