For a knotted graph in S3 we define the vertex constant group, a quotient of the fundamental group of the complement. For planar graphs the group is cyclic. For graphs with periodic symmetry the group is related to the fundamental group of the branched cover of S3 branched over knots contained in the quotient of the graph under the symmetry. These tools are used to demonstrate that a large fami...

For a knotted graph in S* we define the vertex constant group, a quotient of the fundamental group of the complement. For planar graphs the group is cyclic. For graphs with periodic symmetry the group is related to the fundamental group of the branched cover of S3 branched over knots contained in the quotient of the graph under the symmetry. These tools are used to demonstrate that a large fami...

A finite graph Γ is said to be G-symmetric if G is a group of automorphisms of Γ acting transitively on the ordered pairs of adjacent vertices of Γ. In most cases, the group G acts imprimitively on the vertices of Γ, that is, the vertex set of Γ admits a nontrivial G-invariant partition B. The purpose of this thesis is to study such graphs, called imprimitive G-symmetric graphs. In the first pa...