Congruence classes of orientable 2-cell embeddings of bouquets of circles and dipoles.
For an ordered set of vertices and a vertex in a connected graph , the representation of with respect to is the -vector = (, , where represents the distance between the vertices and . The set is a resolving set for if distinct vertices of have distinct representations with respect to . A resolving set for containing a minimum number of vertices is a basis for . The dimension is the number of vertices in a basis for . A resolving set of is connected if the subgraph...
We compute the Coxeter polynomial of a family of Salem trees, and also the limit of the spectral radii of their Coxeter transformations as the number of their vertices tends to infinity. We also prove that if z is a root of multiplicities for the Coxeter polynomials of the trees respectively, then z is a root for the Coxeter polynomial of their join, of multiplicity at least where .