# KPPY 70

Jack Koolen, Jaeyoung Yang, Alexander Gavrilyuk, Jongyoon Hyun, Andreas Holmsen

May 23, 2015. 11am - 5:30pm at Postech

Schedule | ||

11:00am - 11:50 |
Jack Koolen USTC | On graphs with three distinct eigenvalues |

12pm - 1:25 | Lunch | |

1:30 - 2:20 |
Jaeyoung Yang POSTECH | A structure theory for graphs with fixed smallest eigenvalue |

2:30 - 3:20 |
Alexander Gavrilyuk N.N. Krasovsky Institute of Mathematics and Mechanics of Ural Branch of Russian Academy of Sciences, Russia | On Deza circulants |

3:30 - 4:20 |
Jongyoon Hyun Ewha Woman's University | Characterization of weakly regular $p$-ary bent functions for construction of strongly regular graphs |

4:30 - 5:20 |
Andreas Holmsen KAIST | Orthogonal colorings of the sphere |

## Abstracts

Jack Koolen

On graphs with three distinct eigenvalues

On graphs with three distinct eigenvalues

Regular connected graphs with three distinct eigenvalues or equivalently strongly regular graphs have had a lot of consideration in the literature. The case of non-regular graphs has had much less attention. In this talk I will give some
recent results on these graphs.
This is based on joint work with Ximing Cheng, Gary Greaves and Alexander Gavrilyuk

Jaeyoung Yang

A structure theory for graphs with fixed smallest eigenvalue

A structure theory for graphs with fixed smallest eigenvalue

Let $\tilde{K}_{2t}$ be the graph on $2t+1$ vertices consisting of a complete graph $K_{2t}$ and a vertex which is adjacent to exactly $t$ vertices of the $K_{2t}$. It is easy to show that if the smallest eigenvalue of a graph $G$ is at least a fixed real number $\lambda$, then there exists a positive integer $t$ such that $G$ does not contain both $\tilde{K}_{2t}$ and
$K_{1,t}$. In this talk, we obtain a structure theory for graphs with fixed smallest eigenvalue by using this property.

Alexander Gavrilyuk

On Deza circulants

On Deza circulants

A connected regular graph $\Gamma$ is a Deza graph, if there exist integers $\lambda$ and $\mu$ such that any two distinct vertices of $\Gamma$ have either $\lambda$ or $\mu$ common neighbours.
A circulant is a graph that admits a cyclic group of automorphisms, i.e., it is a Cayley graph of a cyclic group.
In this talk, we report on our attempt (in progress) to classify circulants that are Deza graphs.
This is joint work with Sergey Goryainov and Leonid Shalaginov.

Jongyoon Hyun

Characterization of weakly regular $p$-ary bent functions for construction of strongly regular graphs

Characterization of weakly regular $p$-ary bent functions for construction of strongly regular graphs

We characterize the weakly regular $p$-ary bent functions (from
$\mathbb{Z}^n_p$ to $\mathbb{Z}_p$) for constructing strongly
regular graphs, where $p$ is a prime number. Our work is motivated by interesting results
by Chee et al and Tan et al. We, however, use completely different approach, and
our parameters of strongly regular graphs include their parameters.

Andreas Holmsen

Orthogonal colorings of the sphere

Orthogonal colorings of the sphere

An orthogonal coloring of the two-dimensional sphere is a partition into parts such that no part contains a pair of orthogonal points. It is a well-known result that an orthogonal coloring of the 2-sphere requires at least four parts, and orthogonal colorings with exactly four parts can easily be constructed from a regular octahedron centered at the origin. An intriguing question is whether or not every orthogonal 4-coloring of the 2-sphere is such an octahedral coloring. I will show that if every color class has a non-empty interior, then the coloring is octahedral. Some consequences and related results are also given. This is joint work with Seunghun Lee.