X iv :h ep -p h/ 05 11 14 7v 1 1 1 N ov 2 00 5 IPPP/05/67 DCPT/05/134 Studying unquenching effects in QCD with Dyson-Schwinger equations ∗ Christian S. Fischer, Reinhard Alkofer, Wolfgang Cassing, Felipe Llanes-Estrada, Peter Watson, IPPP, University of Durham, Durham DH1 3LE, U.K. Institute of Physics, Graz University, Universitätsplatz 5, A-8010 Graz, Austria Institute for Theoretical Physics...

Albertson conjectured that if a graph G has chromatic number r, then the crossing number of G is at least as large as the crossing number of Kr, the complete graph on r vertices. Albertson, Cranston, and Fox verified the conjecture for r 6 12. In this paper we prove it for r 6 16. Dedicated to the memory of Michael O. Albertson.

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrödinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index-an index characterizing the importance of eigenvalues close to zero. This index accounts for th...