18 research outputs found
Random Graphs: From Paul Erdős to the Internet
Paul Erdős, one of the greatest mathematicians of the twentieth century, was a champion of applications of probabilistic methods in many areas of mathematics, such as a graph theory, combinatorics and number theory. He also, almost fifty years ago, jointly with another great Hungarian mathematician Alfred Rényi, laid out foundation of the theory of random graphs: the theory which studies how large and complex systems evolve when randomness of the relations between their elements is incurred. In my talk I will sketch the long journey of this theory from the pioneering Erdős era to modern attempts to model properties of large real world networks which grow unpredictably, including the Internet, World Wide Web (WWW), peer-to-peer, social, neural and metabolic networks.https://egrove.olemiss.edu/math_dalrymple/1006/thumbnail.jp
The 1-2-3 Conjecture for Hypergraphs
A weighting of the edges of a hypergraph is called vertex-coloring if the
weighted degrees of the vertices yield a proper coloring of the graph, i.e.,
every edge contains at least two vertices with different weighted degrees. In
this paper we show that such a weighting is possible from the weight set
{1,2,...,r+1} for all hypergraphs with maximum edge size r>3 and not containing
edges solely consisting of identical vertices. The number r+1 is best possible
for this statement.
Further, the weight set {1,2,3,4,5} is sufficient for all hypergraphs with
maximum edge size 3, up to some trivial exceptions.Comment: 12 page
Properties of stochastic Kronecker graphs
The stochastic Kronecker graph model introduced by Leskovec et al. is a
random graph with vertex set , where two vertices and
are connected with probability
independently of the presence or absence of any other edge, for fixed
parameters . They have shown empirically that the
degree sequence resembles a power law degree distribution. In this paper we
show that the stochastic Kronecker graph a.a.s. does not feature a power law
degree distribution for any parameters . In addition,
we analyze the number of subgraphs present in the stochastic Kronecker graph
and study the typical neighborhood of any given vertex.Comment: 37 pages, 2 figure
An iterative approach to graph irregularity strength
AbstractAn assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal edge weight, minimized over all irregular assignments, and is set to infinity if no such assignment is possible. In this paper, we take an iterative approach to calculating the irregularity strength of a graph. In particular, we develop a new algorithm that determines the exact value s(T) for trees T in which every two vertices of degree not equal to two are at distance at least eight