KPPY 79

A real Bott manifold is a closed smooth manifold obtained as the total space of an iterated circle-bundles starting with a point, where each circle-bundle is the projectivization of the Whitney sum of two real line bundles. A 2-dimensional torus and a Klein bottle provide examples of real Bott manifolds. These manifolds are quite interesting objects from the combinatorial view point because they can be assigned to acyclic digraphs. In this talk, we shall show that the diffeomorphism types of real Bott manifolds can be completely characterized in terms of graph operations on directed acyclic graphs including local complementation. Using this combinatorial interpretation, we discuss several properties of real Bott manifolds, and introduce recent works on this topic.

A nowhere-zero $k$-flow in $G=(V,E)$ is a pair $(\vec{G},\psi),$ where $\vec{G}$ is an orientation of $G$ and $\psi : E\to \{-k+1,\ldots,-1,1,\ldots,k-1\}$ satisfies Kirchhoff's Law of flow conservation:\[ \sum_{\{ e\in E \, : \, h(e)=v \} } \psi(e) - \sum_{\{ e\in E \, : \, t(e)=v \} } \psi(e) = 0 \, \, \, \mbox{for all $v \in V.$}\] We characterize constructively the class of graphs that do not allow a nowhere-zero 3-flow.

In this talk, we will show that the $2$-clique extension of the $( t \times t)$-grid is determined by its spectrum if $t$ is large enough. In order to show this, we use Hoffman graphs as our main tools. Hoffman graphs were introduced by Woo and Neumaier (1995) and were derived from Hoffman's idea in his 1977 paper. Later, Kim, Koolen and Yang (2016) and Koolen, Yang, Y. (201?) developed further the theory of Hoffman graphs and proved that if $t$ is very large, the 2-clique extension of the $(t \times t)$-grid is a $2$-fat ,,-line Hoffman graph. Based on this fact, we show that the $2$-clique extension of the $( t \times t)$-grid is determined by its spectrum if $t$ is large enough.

(This is joint work with Aida Abiad (Maastricht) and Jack Koolen (USTC).)

Graphs with at most three distinct eigenvalues are well-studied. A main eigenvalue of a graph is an eigenvalue with an eigenvector not orthogonal to the all one vector. A plain eigenvalue is an eigenvalue with an eigenvector orthogonal to the all-one vector. Note that usually a bi-regular graph with three distinct eigenvalues has two main and two plain eigenvalues. In this talk I will consider the class of graphs with two main and two plain eigenvalues. A graph in the switching class of a non-trivial regular two-graph has this property. It was shown by Van Dam, K. and Xia that such a graph can have as many distinct valencies as you prefer.

This is joint work with Sakander Hayat and Muhammad Javaid.

We provide general criteria for orthogonal arrays and $t$-designs on equitable partitions of a hypergraph by exploring harmonic distributions. Generalized harmonic weight enumerators for complex-valued functions of hypergraph are introduced and applied to eigenfunctions of the adjacency matrix of hypergraph. Using this, expressions for harmomic distributions are established for every cell of an equitable partition of hapergraph. Moreover, for any given cell in the partition, the strength of the cell as an orthogonal array is explicitly expressed, and also a characterization of a $t$-design of that cell is established.

This is joint work with Hyun Kwang Kim, Jongyoon Hyun.