扇图的 Terwilliger 代数

Terwilliger 代数是刻画距离正则图的局部结构的重要工具，但利用 Terwilliger 代数刻画一般图的结构的研究较少。文章研究扇图的 Terwilliger 代数。首先证明了 Terwilliger 代数在图的自同构群的作用下保持同构，并给出扇图的自同构群；其次完全刻画了其关于任意基本点的 Terwilliger 代数的不可约模的结构；最后得到扇图的 Terwilliger 代数与其自同构群的点稳定化子的中心化子代数同构的充要条件。

Terwilliger algebra is an important tool for characterizing the local structure of distance-regular graphs, but there are few researches on characterizing the structure of general graphs by the Terwilliger algebra. In this paper, the Terwilliger algebras of a fan graph is studied. Firstly, it is proved that Terwilliger algebras keep isomorphism under the action of automorphism group on graph and the automorphism group of the fan graph is given. Secondly, the structure of irreducible modules for each of Terwilliger algebras of the fan graphs is determined. As a result, a necessary and sufficient condition for the isomorphism between the Terwilliger algebra for a fixed vertex and the centralizer algebra of stabilizer of the automorphism group of the fan graph is obtained.