Antipodal covers of strongly regular graphs

Abstract

Antipodal covers of strongly regular graphs which are not necessarily distance-regular are studied. The structure of short cycles in an antipodal cover is considered. In most cases, this provides a tool to determine if a strongly regular graph has an antipodal cover. In these cases, covers cannot be distance-regular except when they cover a complete bipartite graph. A relationship between antipodal covers of a graph and its line graph is investigated. Finally, antipodal covers of complete bipartite graphs and their line graphs are characterized in terms of weak resolvable transversal designs which are, in the case of maximal covering index, equivalent to affine planes with a parallel class deleted. This generalizes Drake's and Gardiner's characterization of distance-regular antipodal covers of complete bipartite graphs. Bipartite antipodal distance-regular graphs with odd diameter are characterized.