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JCT
2007
13 years 8 months ago
2007
We give a new upper bound for the rectilinear crossing number cr(n) of the complete geometric graph Kn. We prove that cr(n) ≤ 0.380559 ¡n 4 ¢ + Θ(n3 ) by means of a new const...
JCT
2007
13 years 8 months ago
2007
In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan’s “lost” notebook, we obtain weighted forms of Euler’s theorem on part...
JCT
2007
13 years 8 months ago
2007
We prove a number of relations between the number of cliques of a graph G and the largest eigenvalue (G) of its adjacency matrix. In particular, writing ks (G) for the number of s...
JCT
2007
13 years 8 months ago
2007
The preferential attachment graph Gm(n) is a random graph formed by adding a new vertex at each time step, with m edges which point to vertices selected at random with probability...
JCT
2007
13 years 8 months ago
2007
An induced subgraph is called homogeneous if it is either a clique or an independent set. Let hom(G) denote the size of the largest homogeneous subgraph of a graph G. In this shor...