Some large graphs with given degree and diameter

This paper considers the (Δ, D) problem: to maximize the order of graphs with given maximum degree Δ and diameter D, of importance for its implications in the design of interconnection networks. Two cubic graphs of diameters 5 and 8 and orders 70 and 286, respectively, and a graph of degree 5, diameter 3 and order 66 are presented, which improve the previously known orders for these values of Δ and D. By its construction, these graphs have a large automorphism group.