An Introduction to Algebraic Graph Theory
Spectrum, Terwilliger Algebra and its Generalization
Hiroshi SUZUKI∗ Department of Mathematics and Computer Science, International Christian University, Mitaka, Tokyo 181-8585, JAPAN
In algebraic graph theory, we study structures of graphs by properties of algebraic objects associated to them. The first one to look at is the spectrum of a graph. But it turns out that many basic properties cannot be retrieved from its spectrum. In 90s Terwilliger defined the subconstituent algebra of a graph, which is also called Terwilliger algebra. Studying the irreducible modules of Terwilliger algebra makes it possible to have closer look at the structure of a graph and it was successfully applied mainly to Q-polynomial distanceregular graphs. In this talk, we define the Terwiliger algebra of a polynomial space consisting of a Hermitian matrix A ∈ Matn (C) and an orthogonal direct sum decomposition of the Hermitian space V = C n into subspaces V0 , V1 , . . . , Vt satisfying AVi ⊂ Vi−1 + Vi + Vi+1 , for all i ∈ {0, 1, . . . , t} with V−1 = Vt+1 = 0. We discuss possible areas of applications of this algebra.
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Electric mail : hsuzuki@icu.ac.jp