Classification of Infinite Coxeter Graphs Relative to the Value of the Index in the Interval (√√5 + 2; 3/√2]

Abstract

The structure of infinite Coxeter graphs whose largest eigenvalue belongs to the interval from √√5 + 2 to 3 3/√2] is investigated. In particular, such a graph is a tree, can have at most one label greater than 3 on its edges and such label does not exceed 6, can have only vertices with degree strictly less than 5, and among edges which are incident with vertex with degree 4 can be only one that is not incident with leaf. A number of other properties are also given for infinite Coxeter graphs with largest eigenvalue in the specified interval.