Jie Xue (ECNU)

Title: On the eigenvalues of A -spectra of graphs

Abstract: Let G be a graph with adjacency matrix A(G) and degree diagonal matrix

D(G). For any real 2 [0; 1], Nikiforov de ned the matrix A (G) as

A (G) = D(G) + (1 )A(G):

We study the eigenvalues of the A matrix. We show that k(A (G)) n 1

for 2 k n and the extremal graphs are characterized. We also prove that some

graphs are determined by their A spectra. This is based on joint work with Huiqiu

Lin, Xiaogang Liu and Jinlong Shu.

Sakander Hayat, Muhammad Riaz (USTC)

Title: Co-edge-regular graphs which are cospectral with the s-clique extension of the

square grid graphs

Abstract: In this talk, we will discuss our recent result which states that any co-edge-

regular graph which is cospectral with the s-clique extension of the (tt)-grid is the

s-clique extension of the (t t)-grid, if t 100s4. By applying results of Gavrilyuk

and Koolen, this implies that the Grassmann graph Jq(2D;D) is determined by its

intersection array as a distance-regular graph, if D > 20 and if q 9, then D > 7.

This is joint work with Jack Koolen.

Shuang-Dong Li (AHU)

Title: The Terwilliger algebra of a tree

Abstract: Let be a nite connected simple graph. Let X denote the vertex set of

and V =Lx2X Cx the standard module, i.e., the vector space for which X is an

orthonormal basis. Fix a vertex x0 2 X and let Xi be the set of vertices that have

distance i from x0. Then the standard module V is decomposed into the orthogonal

sum V =LDi=0 V i , where V i =Lx2Xi Cx. The Terwilliger algebra T of is by

de nition the subalgebra of End(V ) generated by the adjacency matrix A of and

the orthogonal projections Ei : V ! V i , 0 i D. Let G be the automorphism

group of and H the stabilizer in G of the base vertex x0: G = Aut(), H = Gx0 .

Then it is easy to see that T is contained in the centralizer algebra of H, i.e., each

element of T commutes with the action of every element of H: T HomH(V; V ).

In this talk, we discuss the Terwilliger algebra of a tree. Precisely speaking, we

assume is a rooted tree with x0 the root and we let T be the Terwilliger algebra

of with respect to x0. We show: (1) T = HomH(V; V ), i.e., T coincides with the

centralizer algebra of H. (2) The T-module V determines the rooted tree up to

isomorphism. In particular, T = End(V ) holds if and only if the rooted tree does

not have any symmetry, i.e., H = 1. Note that the Terwilliger algebra as an abstract

algebra cannot determine the rooted tree up to isomorphism.

This talk is based on joint work with Jing Xu, Masoud Karimi, Yizheng Fan and

Tatsuro Ito. We acknowledge that Jack Koolen conjectured: For almost all nite

connected simple graphs, T = End(V ) holds. This conjecture motivated our study

on the Terwilliger algebra of a tree.

Sergey Goryainov (SJTU)

Title: Eigenfunctions of the Star graphs

Abstract: The Star graph Sn, n > 2, is the Cayley graph on the symmetric group

Symn generated by the set of transpositions f(12); (13); :::; (1n)g. We present a family

of eigenfunctions of Sn corresponding to the eigenvalue n m 1 for any positive

integers m and n, where 2m < n and n > 3, and show a connection between this

family and eigenvectors of the Jucys-Murphy element Jn.

This is joint work with V. Kabanov, E. Konstantinova, L. Shalaginov and A. Va-

lyuzhenich