New large graphs with given degree and diameter six

In this article, a new method is proposed for obtaining large-diameter 6 graphs by replacing some vertices of a Moore bipartite diameter 6 graph by complete K h graphs. These complete graphs are joined to the remaining nonmodified graph and to each other by means of new edges. More precisely, these new edges are those of a bipartite diameter 3 graph, together with some appropriate extra edges. The degree and the diameter of the graph so constructed coincide with those of the original one. The construction put forward here gives rise to some of the largest known diameter 6 graphs.

On large (Δ, 6)-Graphs

Martí¹

2005

Two new families of large compound graphs

Martí¹,

Miller

2006

A question of special interest in graph theory is the design of large graphs. Specifically, we want to find constructions of graphs with order as large as possible for a given degree and diameter D. Two generalizations of two large compound graphs are proposed in this article. Three particular cases of these families of graphs presented here allow us to improve the order for the entries (15, 7), (13, 10), and (15, 10) in the table of the largest known ( , D)-graphs.

New largest known graphs of diameter 6

Pineda-Villavicencio

Gómez²,

Miller

et al. 2008

In the pursuit of obtaining largest graphs of given maximum degree and diameter D, many construction techniques have been developed. Compounding of graphs is one such technique. In this article, by means of the compounding of complete graphs into a bipartite Moore graph of diameter 6, we obtain a family of large graphs of the same diameter. For maximum degrees = 5, 6, 9, 12, and 14, members of this family constitute the largest known graphs of diameter 6.