Circular embeddings of planar graphs in nonspherical surfaces

AbstractWe show that every 3-connected planar graph has a circular embedding in some nonspherical surface. More generally, we characterize those planar graphs that have a 2-representative embedding in some nonspherical surface

Spectra and eigenspaces of arbitrary lifts of graphs

We describe, in a very explicit way, a method for determining the spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not)

Large vertex-transitive graphs of diameter 2 from incidence graphs of biaffine planes

Under mild restrictions, we characterize all ways in which an incidence graph of a biaffine plane over a finite field can be extended to a vertex-transitive graph of diameter 2 and a given degree with a comparatively large number of vertices

Maximum genus embeddings of Steiner triple systems

We prove that for n>3 every STS(n) has both an orientable and a nonorientable embedding in which the triples of the STS(n) appear as triangular faces and there is just one additional large face. We also obtain detailed results about the possible automorphisms of such embeddings

Biembeddings of Steiner triple systems in orientable pseudosurfaces with one pinch point

We prove that for all n ≡ 13 or 37 (mod 72), there exists a biembedding of a pair of Steiner triple systems of order n in an orientable pseudosurface having precisely one regular pinch point of multiplicity 2

Small surface trades in triangular embeddings

We enumerate all possible trades which involve up to six faces of the face set of a triangular embedding of a simple connected graph. These are classified by the underlying combinatorial trade on the associated block design, and by the geometrical arrangement of the faces necessary to avoid creation of a pseudosurface in the trading operation. The relationship of each of these trades to surface orientability is also established