Showing papers on "Connectivity published in 1981"
TL;DR: In this article, the authors considered the class of stochastic matrices M generated in the following way from graphs: if G is an undirected connected graph on n vertices with adjacency matrix A, we form M from A by dividing the entries in each row of A by their row sum.
111 citations
TL;DR: It is shown that the persistence of a graph of diameter d is the minimum over all pairs of nonadjacent points of the maximum number of disjoint paths of length at most d joining them.
Abstract: It is well known that the maximum connectivity k of a graph G with p points and q lines is given by [2 q/p]. This is restated in two useful alternative forms which minimize q given p and k, and which maximize p in terms of q and k. We define the persistence of a graph as the smallest number of points whose removal increases the diameter. It is shown that the persistence of a graph of diameter d is the minimum over all pairs of nonadjacent points of the maximum number of disjoint paths of length at most d joining them. A similar result is obtained for line-persistence and it is shown that these invariants are independent of each other.
55 citations
01 Jan 1981
TL;DR: In this article, the realizations of a given degree sequence, and the way in which they are related by switchings are investigated in the context of simple graphs, multigraphs and pseudographs.
Abstract: This paper investigates realizations of a given degree sequence, and the way in which they are related by switchings The results are given in the context of simple graphs, multigraphs and pseudographs We show that we can transform any connected graph to any other connected graph of the same degree sequence, by switchings wich are constrained to connected graphs This is done for certain labelled graphs, the result for unlabelled graphs following as a corollary The results are then extended to infinite degree sequences
44 citations
01 Jan 1981
TL;DR: Erdős as discussed by the authors discusses problems connected with Ramsey's theorem and its generalisations, here I of course can not avoid overlap with previous papers, but this paper contains extensive references to my previous papers.
Abstract: Recently I published several papers on finite and infinite combinatorial problems. I will try to make the overlap with this paper as small as possible ; as a result I have to omit some of my most interesting problems, but first of all I give some references to my older papers where these questions have been discussed P. Erdős, Old and new problems in combinatorial analysis and graph theory, Secondference). This paper contains extensive references to my previous papers. P. Erdős, Combinatorial problems which I would most like to see solved, will soon appear in the new Hungarian periodical Combinatorics. For applications of probabilistic methods to combinatorial analysis see our book, First I discuss problems connected with Ramsey's theorem and its generalisations, here I of course can not avoid overlap with previous papers. r(nl,. . .,nk) is the smallest integer for which if one colors the edges of K(r(nl,. . .,nk)) by k colors (K(t)
25 citations
TL;DR: In this paper, a new definition of matroid connectivity is introduced and its properties are investigated and compared with Tutte connectivity, and it is shown that the Whitney connectivity of a polygon matroid of a graph is the same as the vertex connectivity of the graph provided the graph is connected.
Abstract: A new definition of matroid connectivity is introduced and its properties are investigated in this paper. Vertex connectivity of graphs is expressed in an algebraic form and generalized to matroids. This generalized connectivity is called the Whitney connectivity of matroids. It is shown that the Whitney connectivity of the polygon matroid of a graph is the same as the vertex connectivity of the graph provided the graph is connected. Various properties of Whitney matroid connectivity and comparison with Tutte connectivity are also examined.
10 citations
01 Jan 1981
TL;DR: In this paper, the authors characterize potentially connected integer-pair sequences, where the edges of a pseudograph, multigraph or graph G are labeled 1,2, …, q, and for the sth edge (u,v) of G, cs=(as, bs) where as, b s are the degrees of u and v.
Abstract: Given a pseudograph, multigraph or graph G, we can associate with it a sequence of unordered integer-pairs SG=(c1, c2, …, cq), where q=|E(G)|, constructed as follows: If the edges of G are labelled 1,2, …, q, then for the sth edge (u,v) of G, cs=(as, bs) where as, bs are the degrees of u and v. An integer-pair sequence S is said to be graphic if there exists a graph G for with SG=S. In this paper we characterize potentially connected integer-pair sequences.
9 citations