Results 1 
3 of
3
On the Evolution of Random Graphs
 PUBLICATION OF THE MATHEMATICAL INSTITUTE OF THE HUNGARIAN ACADEMY OF SCIENCES
, 1960
"... his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_ ..."
Abstract

Cited by 2043 (8 self)
 Add to MetaCart
his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_
Graphs with a prescribed adjacency property
, 1992
"... A graph G is said to have property P(m,n,k) if for any set of m + n distinct vertices of G there are at least k other vertices, each of which is adjacent to the first m vertices of the set but not adjacent to any of the latter n vertices. The problem that arises is that of characterizing graphs havi ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
A graph G is said to have property P(m,n,k) if for any set of m + n distinct vertices of G there are at least k other vertices, each of which is adjacent to the first m vertices of the set but not adjacent to any of the latter n vertices. The problem that arises is that of characterizing graphs having property P(m,n,k). In this paper, we present properties of graphs satisfying the adjacency property. In addition, for small m and n we show that all sufficiently large Paley graphs satisfy P(m,n,k).
On graphs satisfying a strong adjacency property
, 1993
"... Let m and n be nonnegative integers and k be a positive integer. A graph G is said to have property P*(m,n,k) if for any set of m + n distinct vertices of G there are exactly k other vertices, each of which is adjacent to the first m vertices of the set but not adjacent to any of the latter n vertic ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Let m and n be nonnegative integers and k be a positive integer. A graph G is said to have property P*(m,n,k) if for any set of m + n distinct vertices of G there are exactly k other vertices, each of which is adjacent to the first m vertices of the set but not adjacent to any of the latter n vertices. The case n = 0 is, of course, a generalization of the property in the Friendship Theorem. In this paper we show that, for m = n = 1, graphs with this property are the ( (k+t)\l socalled strongly regular graphs with parameters t,k+t, tl, t) for some positive integer t. In particular, we show the existence of such graphs. For m ~ 1, n ~ 1, and m + n ~ 3, we show that, there is no graph having property P*(m,n,k), for any positive integer k.