In our previous paper [3], we introduced the notion of weighted Coxeter graph and associated to it a certain generalization of the standard geometric representation of a Coxeter group. In the present paper, we continue to study this subject. When we put signs on the edges of a simply laced Coxeter graph, we obtain a certain quotient of the corresponding to this graph Coxeter group. The natural question is how many different quotient-groups can be obtained this way from a given graph. In the present paper, we give a partial solution to this problem dealing with line graphs with a certain natural restriction.

Coxeter groups, Coxeter graphs, geometric representation, signed graphs, weighted graphs, balanced functions on graphs, line graphs.