4 research outputs found
On large bipartite graphs of diameter 3
We consider the bipartite version of the {\it degree/diameter problem},
namely, given natural numbers and , find the maximum number
of vertices in a bipartite graph of maximum degree and diameter
. In this context, the bipartite Moore bound \M^b(d,D) represents a
general upper bound for . Bipartite graphs of order \M^b(d,D) are
very rare, and determining still remains an open problem for most
pairs.
This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on
bipartite -graphs (that is, bipartite graphs of order \M^b(d,D)-4)
was carried out. Here we first present some structural properties of bipartite
-graphs, and later prove there are no bipartite -graphs.
This result implies that the known bipartite -graph is optimal, and
therefore . Our approach also bears a proof of the uniqueness of
the known bipartite -graph, and the non-existence of bipartite
-graphs.
In addition, we discover three new largest known bipartite (and also
vertex-transitive) graphs of degree 11, diameter 3 and order 190, result which
improves by 4 vertices the previous lower bound for
Degree/diameter problem for mixed graphs
The Degree/diameter problem asks for the largest graphs given diameter and maximum degree. This problem has been extensively studied both for directed and undirected graphs, ando also for special classes of graphs. In this work we present the state of art of the degree/diameter problem for mixed graphs