Semi-affine Coxeter-dynkin Graphs and G ⊆ Su 2 (c)

The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types. Semi-affine graphs. It is profitable to treat the so-called Coxeter-Dynkin diagrams as graphs. A classification of finite graphs with an adjacency matrix having 2 as the largest eigenvalue is made in a paper of John Smith [JHS]. It is in a combinatorial context and no reference is made to Coxeter-Dynkin diagrams there. This maximal eigenvalue property is a defining property of the affine diagrams. What is introduced in this note is a more weakly constrained graph, and we examine its eigenvalues and interpret the rational functions which arise in terms of my correspondence [M1,K]. Since these semi-affine graphs do not have symmetrizable matrices, this appears to imply a connection with singularities rather than Lie algebras. Here we shall deal only with those of type A, D, and E. Undirected edges are treated as a pair of edges directed in opposing directions as in [FM,M1,M2]. By so doing, we can introduce the semi-affine graph which may be defined in terms of a graph of finite type with an additional edge (two for A-type) directed toward the affine node; equivalently it may be defined as an affine graph with any undirected edge connecting the affine node replaced by an directed edge directed toward the affine node. This is done by removing one of the two opposed directed edges. Effectively the semi-affine graph generalizes both the affine and finite type graph since the affine node acts as a sink, and when weighted at the nodes, it satisfies the same constraints as the affine graph except the constraint imposed by the additional directed edge(s) in the affine graph. Note that weighting the extra node with zero yields the same constraints as the finite type graph. In some sense the semi-affine graph lies intermediate between the finite and affine graphs yet generalizes both. Partially supported by NSERC and FCAR grants. Typeset by AMS-TEX 1