Graphs Without Dead Ends

We consider graphsGwhich are antipodal in the sense that the distance function ofG,rooted at any vertex, has a unique relative maximum. These graphs can be characterised as a certain type of 2-cover of their natural quotient, obtained by identifying each vertex with its antipode. We use this characterisation to discuss various properties of antipodal graphs, e.g. their degrees, girth, and automorphisms.