136 research outputs found
Convexities related to path properties on graphs; a unified approach
Path properties, such as 'geodesic', 'induced', 'all paths' define a convexity on a connected graph. The general notion of path property, introduced in this paper, gives rise to a comprehensive survey of results obtained by different authors for a variety of path properties, together with a number of new results. We pay special attention to convexities defined by path properties on graph products and the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants, such as clique numbers and other graph properties.
How likely is an i.i.d. degree sequence to be graphical?
Given i.i.d. positive integer valued random variables D_1,...,D_n, one can
ask whether there is a simple graph on n vertices so that the degrees of the
vertices are D_1,...,D_n. We give sufficient conditions on the distribution of
D_i for the probability that this be the case to be asymptotically 0, {1/2} or
strictly between 0 and {1/2}. These conditions roughly correspond to whether
the limit of nP(D_i\geq n) is infinite, zero or strictly positive and finite.
This paper is motivated by the problem of modeling large communications
networks by random graphs.Comment: Published at http://dx.doi.org/10.1214/105051604000000693 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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