Book Chapter10.1007/978-94-010-9910-3_21
Random Graph Theorems
Geoffrey Grimmett
- 01 Jan 1977
- pp 203-209
17
TL;DR: In this article, the authors studied the chromatic number of a random graph on n vertices in which each edge is present with a prescribed probability p independently of the presence or absence of the other edges.
read more
Abstract: In an earlier paper [3] McDiarmid and I studied the chromatic number of a random graph on n vertices in which each edge is present with a prescribed probability p independently of the presence or absence of the other edges. Such random graphs differ from those which Erdos and Renyi considered in their paper [1] on the evolution of random graphs, for they were interested in graphs on n vertices with N(n) edges where N is a prescribed function and each N-subset of the set of possible edges occurs with equal probability. They noted that there is no essential difference between the two approaches.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
A Theory of Network Localization
James Aspnes,Tolga Eren,David K. Goldenberg,A.S. Morse,Walter Whiteley,Yang Yang,Brian D. O. Anderson,Peter N. Belhumeur +7 more
TL;DR: This paper constructs grounded graphs to model network localization and applies graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks, and further study the computational complexity of network localization.
Rigidity, computation, and randomization in network localization
Tolga Eren,O.K. Goldenberg,Walter Whiteley,Yang Yang,A.S. Morse,Brian D. O. Anderson,Peter N. Belhumeur +6 more
- 07 Mar 2004
TL;DR: This work provides a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors and constructs grounded graphs to model network localization.
A review of random graphs
TL;DR: This paper presents a review of results dealing with structural properties and numerical characteristics of random edge-subgraphs of complete graphs, with the connectedness of random graphs, as well as with random trees and forests.
90
Graph theoretical insights into evolution of multidomain proteins
Teresa M. Przytycka,George B. Davis,Nan Song,Dannie Durand +3 more
- 14 May 2005
TL;DR: Connections between properties of the domain overlap graph and certain variants of Dollo parsimony models are demonstrated, indicating that independent merges of domain pairs are not uncommon in large superfamilies.
Modularity and anti-modularity in networks with arbitrary degree distribution
Arend Hintze,Christoph Adami +1 more
TL;DR: In this article, the authors study how a network's growth parameters impact the distribution of edges in the network, how they affect network's modularity, and point out that some parameters will give rise to networks that have the opposite tendency to display anti-modularity.
References
On colouring random graphs
Geoffrey Grimmett,Colin McDiarmid +1 more
- 01 Mar 1975
TL;DR: In this paper, it was shown that the number of vertices in the largest complete subgraph of ωn is, with probability one, the same as in this paper.
406
Enumeration Of Labelled Graphs
TL;DR: The number of connected linear graphs having V vertices labelled 1, …, V and λ (unlabelled) lines is found below and similar formulas are found for graphs in which slings, lines “in parallel,” or both are allowed and for directed graphs with or without slings or parallel lines.
On dichromatic polynomials
TL;DR: In this article, a study of the combinatorial properties of the dichromatic polynomials of graphs, especially those properties theoretically applicable to the recursive calculation of the polynomial, is made.