** Organizers:**Tatsuro Ito (Anhui University) andJack Koolen (University of Science and Technology of China)

** Date:**June 9, 2018 (Saturday)

** Place:**Room 113, H Building, Qingyuan Campus of Anhui University

**10:30- 11:20**Denis Krotov

(Sobolev Institute of Mathematics, Russia, and AHU)

On the equitable partitions of H(12,2) with quotient matrix [[3,9],[7,5]]

**11:30- 13:15**noon break

**13:15 -14:05**Jongyook Park (Wonkwang University, South Korea)

Clique adjacency bound vs Hoffman bound

**14:20- 15:10**Pengli Zhang (Central South University)

The Q-index and connectedness of graphs

**15:40 -16:30**Jack Koolen (USTC)

A characterization of the Grassmann graphs

**16:45 - 17:35**Istv′an Kov′acs

(IAM and FAMNIT, University of Primorska, Slovenia)

On elementary abelian DCI-groups

**Abstracts**

**Denis Krotov (Sobolev Institute of Mathematics, Russia, and AHU)**

**Title: On the equitable partitions of H(12,2) with quotient matrix [[3,9],[7,5]]**

**Abstract:** In 2007, D.G.Fon-Der-Flaass presented a construction of perfect colorings (equitable partitions) of the 12-cube with parameters [[3,9],[7,5]] and showed that these colorings lie on a correlation-immunity bound. However, the number of the equivalence classes of the constructed colorings, as well as the existence of other colorings with the same parameters, remained unknown. We finish the characterization of this class of coloring by analyzing the Fourier transform of a hypothetical coloring. The Fourier transform of such coloring is shown to correspond to a system of 63 4-subsets of the 12-set 1,...,12 that cover all 3-subsets. Such covering system is exhaustive (a minimum covering is known to have 57 quadruples); however, additional properties allow to characterize all possibilities. The work is funded by the RSF grant 18-11-00136.

**Jongyook Park (Wonkwang University, South Korea)**

**Title: Clique adjacency bound vs Hoffman bound**

**Abstract:** In this talk, we will try to show that for any edge-regular graph, the Clique adjacency bound is (always) as good as the Hoffman bound. In order to do so, we will consider the second largest eigenvalues of (edge-regular) graphs. This is joint work with Gary Greaves.

**Pengli Zhang (Central South University)**

**Title: The Q-index and connectedness of graphs**

**Abstract:** A connected graph G is said to be k-connected if it has more than k vertices and remains connected whenever fewer than k vertices are deleted. In this paper, for a connected graph G with sufficiently large order, we present a tight sufficient condition for G with fixed minimum degree to be k-connected based on the Q-index. Our result can be viewed as a spectral counterpart of the corresponding Dirac type condition.

**Jack Koolen (USTC)**

**Title: A characterization of the Grassmann graphs**

**Abstract:** In the 1990’s Metsch showed that the Grassmann graph Jq(n;D) are characterized by their intersection numbers if n _ max{2D+2; 2D+6-q}. In 2005 Van Dam and K. found the twisted Grassmann graphs ? Jq(2D+1;D) with the intersection numbers as Jq(2D+1;D). In this talk I will discuss the Grassmann graphs Jq(2D;D). This is based on joint work with A. Gavrilyuk (PNU).

**Istv′an Kov′acs (IAM and FAMNIT, University of Primorska, Slovenia )**

**Title: On elementary abelian DCI-groups**

**Abstract:**A subset S of a finite group G is called a CI-subset if whenever the Cayley digraph Cay(G; S) is isomorphic to Cay(G; T) for a subset T; it follows that T = S_with some automorphism _ of G. The group G is called a DCI-group if all its subsets are CI-subsets, and it is called a CI-group if all its inverse closed subsets are CIsubsets. In 1978, Babai and Frankl asked the following question: Which are the CI-groups? The problem of determining all CI- and DCI-groups is still wide open, and in my talk I will focus on the special case when G is an elementary abelian group. I give an account on the current status and describe a method using Schur rings over G (in other words, Cayley association schemes defined over G). I will also discuss two recent results in [1,2], which were obtained using the Schur ring method.

[1] Y.-Q. Feng, I. Kov′acs, Elementary abelian groups of rank 5 are DCI-groups, J. Combin. Theory Ser. A 157 (2018) 162–204.

[2] I. Kov′acs, G. Ryabov, CI-property for decomposable Schur rings over an elementary abelian group, submitted preprint (arXiv:1802.04571[math.CO]).