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The Twelfth AHU-USTC Joint Seminar on Algebraic Combinatorics

  发布日期:2017-12-1  浏览量:1034


 

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


Date: December 2, 2017 (Saturday)
Place: Room 113, H Building, Qingyuan Campus of Anhui University


10:30-- 11:20 Edwin van Dam (Tilburg University)
Strongly walk-regular graphs and digraphs


13:15--14:05 Jie Xue (ECNU)
On the eigenvalues of A -spectra of graphs


14:20--15:10 Sakander Hayat, Muhammad Riaz (USTC)
Co-edge-regular graphs which are cospectral with the s-clique extension of the square
grid graphs


15:40--16:30 Shuang-Dong Li (AHU)
The Terwilliger algebra of a tree


16:45--17:35 Sergey Goryainov (SJTU)
Eigenfunctions of the Star graphs
                                        
   

                                                                          Abstracts

 

Edwin van Dam (Tilburg University)
Title: Strongly walk-regular graphs and digraphs

Abstract: A (di-)graph is called strongly `-walk-regular if the number of walks of
length ` from one vertex to another depends only on whether the two vertices are the
same, adjacent, or not adjacent. This generalizes the concept of (directed) strongly
regular graphs and a problem introduced by Ho man. We present several results and
constructions of strongly `-walk-regular graphs and digraphs. Eigenvalue methods
play a crucial role in obtaining these results.
This talk is based on joint work with Gholamreza Omidi.


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

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