**第十六届安大****-****科大代数组合国际研讨会**

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

**时 间****:**December 22, 2018 (Saturday)

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

**10:30 -- 11:20**Ali Mohammadian (AHU)

Graphs with few distinct Laplacian eigenvalues

**13:15 -- 14:05 **Chen Jiyong (SUSTC)

Edge-transitive maps

**14:20 -- 15:10 **Hyein Choi (Pusan National University, South Korea)

Observation of the multiplicity graph of an association scheme

**15:40 -- 16:30**Mahya Ghandehari (University of Delaware, USA)

A new parameter for seriation of nosey data

**16:45 -- 17:35**Rebecca J. Stones (Nankai University)

Symmetries of Partial Latin Rectangles

**摘要**

Graphs with few distinct Laplacian eigenvalues

Ali Mohammadian (AHU)

Graphs with few distinct Laplacian eigenvalues form an interesting class of graphs and possess nice combinatorial properties. In this talk, we present a short survey on such graphs. We provide a characterization of all graphs with four distinct Laplacian eigenvalues which either are bipartite or have exactly one multiple Laplacian eigenvalue. In addition, some examples of interest will be presented.

Edge-transitive maps

Chen Jiyong (SUSTC)

A map is an embedding of a graph on a closed surface. A map is called edge-transitive if its automorphism group acts transitive on its edge set. In this talk, I will introduce some classical results on map theory, especially on edge-transitive maps, and some recent progress of our own work.

Observation of the multiplicity graph of an association scheme

Hyein Choi (Pusan National University, South Korea)

For a finite group G, its character degree set cd(G)={χ(1)|χ∈Irr(G)} provides a lot of information on the structure of G. There are known results about the character degree graphs related to the number of connected components and the maximal number of independent vertices.

We want to observe above results in terms of association schemes. Define the multiplicity graph of an association scheme whose vertex set is the set of all prime divisors of multiplicities and p and q are adjacent if and only if pq divides for some multiplicity. By investigating the multiplicity graphs of association schemes of small order, we observe similarities and differences between multiplicity graphs and character degree graphs. In this talk, we would like to summarize what we have studied on multiplicity graphs and character degree graphs.

A new parameter for seriation of nosey data

Mahya Ghandehari (University of Delaware, USA)

A square symmetric matrix is Robinsonian if entries in its rows and columns are non-decreasing when moving towards the diagonal. A Robinsonian matrix can be viewed as the affinity matrix between objects arranged in linear order, where objects closer together have higher affinity. Adjacency matrices of geometric graphs are special cases of Robinsonian matrices. In this talk, we introduce a new parameter, Γmax, which recognizes Robinsonian matrices that are perturbed by noise. This parameter can therefore be a useful tool in the problem of seriation of noisy data. More precisely, we show that a matrix is Robinsonian exactly when itsΓmax attains zero, and a matrix with small Γmax is close (in the normalized l1-norm) to a Robinsonian matrix.

Moreover, we show that bothΓmax and the Robinsonian approximation can be computed in polynomial time.

This talk is based on a joint work with Jeannette Janssen.

Symmetries of Partial Latin Rectangles

Rebecca J. Stones (Nankai University)

Partial Latin rectangles are a generalization of Latin squares: they are r × s matrices whose cells may be empty or contain an element from symbol set of size n; we forbid repeated symbols in a row or column. The notions of "symmetry" for Latin squares (autotopisms and autoparatopisms) generalize to partial Latin rectangles.

This talk will describe work on the broad question: for which r, s, n, and m does there exist an m-entry r × s partial Latin rectangle with autotopism group isomorphic to H_{1} and autoparatopism group isomorphic to H_{2}?

This is ongoing joint work with Raúl Fálcon, Dani Kotlar, Trent Marbach, and others.