## Posts Tagged ‘JackKoolen’

### Jack Koolen, m-Walk-regular graphs, a generalization of distance-regular graphs

Friday, May 17th, 2013
m-Walk-regular graphs, a generalization of distance-regular graphs
Jack Koolen
Department of Mathematics, POSTECH
2013/05/31 Friday 4PM-5PM
Walk-regular graph were introduced by Godsil and McKay to understand when the characteristic polynomial of a graph in which a vertex is deleted does not depend on which vertex you delete. This notion was generalized to m-walk-regular graphs by Fiol and Garriga in order to understand how close you can come to a distance-regular graph.

We observed that for many results on distance-regular graphs they also hold for 2-walk-regular. In this talk I will give an overview of which results can be generalized to 2-walk-regular graphs, and I also will give many examples of 2,3,4,5,-walk-regular graphs which are not distance-regular. At this moment all 6-walk-regular graphs known are distance-regular.

This is still work in progress and is joint work with M. Camara, E. van Dam and Jongyook Park.

### Jack Koolen, On connectivity problems in distance-regular and strongly regular graphs

Monday, March 26th, 2012
On connectivity problems in distance-regular and strongly regular graphs
Jack Koolen
Department of Mathematics, POSTECH, Pohang, Korea
2012/4/24 Tue 4PM-5PM
In this talk I will discuss two problems of Andries Brouwer.
In the first one he asked whether the minimal number of vertices you need to delete from a strongly regular graph with valency k and intersection numbers λ, μ, in order to disconnect it and such that each resulting component has at least two vertices is 2k-2-λ. We will show that there are strongly where you can use a smaller number of vertices to disconnect it in this way, but we also will give some positive results.
The second question we discuss is how connected a distance-regular graph is far from a fixed vertex.
This is joint work with Sebastian Cioaba and Kijung Kim.

### Special session on graph theory, 2011 Spring Meeting of the Korean Mathematical Society

Saturday, March 26th, 2011
Special Session on Graph Theory – 2011 spring Meeting of the Korean Mathematical Society
April 30, 2011, 9:00-11:40
Asan Science Building (아산이학관), Korea University (고려대), Seoul

Timetable
• 9:00-9:30 Sang-il Oum (엄상일),  KAIST : Rank-width and well-quasi-ordering of skew-symmetric or symmetric matrices
• 9:30-10:00 Sejeong Bang (방세정), Yeungnam University : Geometric distance-regular graphs with smallest eigenvalue -3
• 10:00-10:10 Break
• 10:10-11:40 Mark H. Siggers, Kyungpook National University : The H-colouring Dichotomy through a projective property
• 10:10-10:40 Tommy R. Jensen, Kyungpook National University : On second Hamilton circuits in cubic graphs
• 11:10-11:40 Jack Koolen, POSTECH : Recent progress of distance-regular graphs

Organized by Seog-Jin Kim (Konkuk University) and Sang-il Oum (KAIST).

At 14:00-14:40, there will be an invited talk by Xuding Zhu, Thue choice number of graphs.

Rank-width and well-quasi-ordering of skew-symmetric or symmetric matrices
Sang-il Oum (엄상일)
Department of Mathematical Sciences, KAIST
We prove that every infinite sequence of skew-symmetric or symmetric matrices M1, M2, … over a fixed finite field must have a pair Mi, Mj (i<j) such that that Mi is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in Mj, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour’s theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle’s theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum’s theorem for graphs of bounded rank-width with respect to pivot-minors.

Geometric distance-regular graphs with smallest eigenvalue −3
Sejeong Bang (방세정)
Department of Mathematics, Yeungnam University
A non-complete distance-regular graph Γ is called geometric if there exists a set C of Delsarte cliques such that each edge of Γ lies in a unique clique in C. In this talk, we determine the non-complete distance-regular graphs satisfying max{3,8(a1+1)/3}<k<4a1+10−6c2. To prove this result, we first show by considering non-existence of 4-claws that any non-complete distance-regular graph satisfying max{3,8(a1+1)/3}<k<4a1+10−6c2 is a geometric distance-regular graph with smallest eigenvalue −3. Moreover, we classify the geometric distance-regular graphs with smallest eigenvalue −3. As an application, 7 feasible intersection arrays are ruled out.

The H-colouring Dichotomy through a projective property
Mark H. Siggers
Department of Mathematics, Kyungpook National University
The H-colouring Dichotomy of Hell and Nesetril, proved in 1990, is one of the most quoted results in the field of Graph Homomorphisms. It says that H-coloring, the problem of deciding if a given graph G admits an homomorphism to the fixed graph H, is NP-complete if H contains an odd cycle, and otherwise polynomial time solvable.
In this talk we present a short new proof of this result, recently published, using a new projective property defined for homomorphisms of powers of a graph G onto a graph H.

On second Hamilton circuits in cubic graphs
Tommy R. Jensen
Department of Mathematics, Kyungpook National University
A classical theorem of Cedric Smith guarantees the existence of a second Hamilton circuit other than a given one in any hamiltonian cubic graph. It is an open problem in complexity theory whether the corresponding search problem is polynomially solvable. We observe that a search algorithm, implicit in Bill Tutte’s nonconstructive proof of Smith’s theorem, has exponential running time. We also mention two possible candidates for search algorithms with polynomial complexity.

Recent progress of distance-regular graphs
Jack Koolen
Department of Mathematics, POSTECH
I will talk about recent progress of distance-regular graphs.

(Invited lecture at 2PM)

Thue choice number of graphs
Xuding Zhu
Institute of Mathematics, Zhejiang Normal University, Jinhua, China
A sequence of even length is a repetition if the first half is identical to the second half. A sequence is said to contain a repetition if it has a subsequence which is a repetition. A classical result of Thue says that there is an infinite sequence on 3 symbols which contains no repetition. This result motivated many deep research and challenging problems. One graph concept related to this result is Thue-colouring. A Thue-colouring of a graph G is a mapping which assigns to each vertex of G a colour (a symbol) in such a way that the colour sequence of any path of G contains no repetition. The Thue-chromatic number of a graph is the minimum number of colours needed in a Thue-colouring of G. Thue’s result is equivalent to say that the infinite path has Thue-chromatic number 3. It is also known that the Thue-chromatic number of any tree is at most 4.
Thue-choice number of a graph G is the list version of its Thue-chromatic number, which is the minimum integer k such that if each vertex of G is given k-permissible colours, then there is a Thue-colouring of G using a permissible colour for each vertex. This talk will survey some research related to Thue Theorem and will show that Thue-choice number of paths is at most 4 and Thue choice number of trees are unbounded.

### Jack Koolen, Some topics in spectral graph theory

Tuesday, March 24th, 2009
Some topics in spectral graph theory
Jack Koolen
Department of Mathematics, POSTECH, Pohang, Korea
2009/03/27 Fri 5PM-6PM (Room 2411)
In spectral graph theory one studies the eigenvalues (and spectrum) of the adjacency matrix and how they are related with combinatorial properties of the underlying graph.
In this talk, I will discuss several topics in spectral graph theory.