Diametral broadcast graphs

This paper studies the family of graphs with broadcast time equal to their diameter. The diametral broadcast graph (dbg) problem   is to answer the question whether for a given nn and dd a graph on nn vertices can be constructed whose diameter and broadcast time are equal to dd. This paper presents several dbg constructions. Together, they solve the dbg problem for all the possible combinations of values of nn and dd. We also define the diametral broadcast function     DB(n,d)DB(n,d) as the minimum possible number of edges in a dbg on nn vertices and diameter dd. We describe all the trees on nn vertices with diametral broadcast time. These trees give the exact value for DB(n,d)DB(n,d) when tree based dbg construction is possible. For the general case we give an upper bound on DB(n,d)DB(n,d).