The (maximum receiver-centric) interference of a geometric graph (von (2005)) is studied. It is shown that, with high probability, the following results hold for a set, V, of n points independently and uniformly distributed in the unit d-cube, for constant dimension d: (1) there exists a connected graph with vertex set V that has interference O((log n)^ 1 3 ); (2) no connected graph with vertex set V has interference o((log n)^ 1 4 ); and (3) the minimum spanning tree of @math has interference Theta(( n)^ 1 2 ).
This lower bound is matched by an upper-bound: The upper bound in twod-upper is obtained by selecting @math vertices to act as , connecting the hubs into any connected network and then having each of the remaining nodes connect to its nearest hub. This idea was extended to two and higher dimensions by Halld 'orsson and Tokuyama @cite_5 , by using a special type of @math -net as the set of hubs:
@cite_5
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We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size @? in a d-regular graph on N vertices. For [email protected]?N bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size @? in the graph consisting of N2d disjoint copies of K"d","d. This provides asymptotic evidence for a conjecture of S. We also obtain an analogous result for independent sets of a fixed size in regular graphs, giving asymptotic evidence for a conjecture of J. Kahn. Our bounds on the number of matchings and independent sets of a fixed size are derived from bounds on the partition function (or generating polynomial) for matchings and independent sets.