Merging in Antipodal Distance-Regular Graphs

Abstract

Merging (also called fusion) is studied in antipodal distance-regular graphs. The conditions are determined under which merging the first and the last classes in an antipodal distance-regular graph produces a distance-regular graph. Conversely, given a distance-regular graph with the same intersection array as the merged graph and a certain clique partition, an antipodal distance-regular graph is constructed. This gives us a characterization of a class of antipodal distance-regular graphs with a class of regular near polygons containing a certain spread, which generalizes Brouwer′s characterization of a class of distance-regular graphs of diameter 3 with generalized quadrangles containing a spread.