Regions without complex zeros for chromatic polynomials on graphs with bounded degree

We prove that the chromatic polynomial $P_\mathbb{G}(q)$ of a finite graph $\mathbb{G}$ of maximal degree \D is free of zeros for \card q\ge C^*(\D) with C^*(\D) = \min_{0 This improves results by Sokal (2001) and Borgs (2005). Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.Comment: 14 pages, to appear in Combinatorics, Probability and Computin

Statistical thermodynamic basis in drug-receptor interactions: double annihilation and double decoupling alchemical theories, revisited

Alchemical theory is emerging as a promising tool in the context of molecular dynamics simulations for drug discovery projects. In this theoretical contribution, I revisit the statistical mechanics foundation of non covalent interactions in drug-receptor systems, providing a unifying treatment that encompasses the most important variants in the alchemical approaches, from the seminal Double Annihilation Method by Jorgensen and Ravimohan [W.L. Jorgensen and C. Ravimohan, J. Chem. Phys. 83,3050, 1985], to the Gilson's Double Decoupling Method [M. K. Gilson and J. A. Given and B. L. Bush and J. A. McCammon, Biophys. J. 72, 1047 1997] and the Deng and Roux alchemical theory [Y. Deng and B. Roux, J. Chem. Theory Comput., 2, 1255 2006]. Connections and differences between the various alchemical approaches are highlighted and discussed, and finally placed into the broader context of nonequilibrium thermodynamics.Comment: 25 pages, 4 figure

A correction to a remark in a paper by Procacci and Yuhjtman: new lower bounds for the convergence radius of the virial series

In this note we deduce a new lower bound for the convergence radius of the Virial series of a continuous system of classical particles interacting via a stable and tempered pair potential using the estimates on the Mayer coefficients obtained in the recent paper by Procacci and Yuhjtman (Lett Math Phys 107:31-46, 2017). This corrects the wrongly optimistic lower bound for the same radius claimed (but not proved) in the above cited paper (in Remark 2 below Theorem 1). The lower bound for the convergence radius of the Virial series provided here represents a strong improvement on the classical estimate given by Lebowitz and Penrose in 1964.Comment: To appear in JS

The analyticity region of the hard sphere gas. Improved bounds

We find an improved estimate of the radius of analyticity of the pressure of the hard-sphere gas in $d$ dimensions. The estimates are determined by the volume of multidimensional regions that can be numerically computed. For $d=2$, for instance, our estimate is about 40% larger than the classical one.Comment: 4 pages, to appear in Journal of Statistical Physic

Properly coloured copies and rainbow copies of large graphs with small maximum degree

Let G be a graph on n vertices with maximum degree D. We use the Lov\'asz local lemma to show the following two results about colourings c of the edges of the complete graph K_n. If for each vertex v of K_n the colouring c assigns each colour to at most (n-2)/22.4D^2 edges emanating from v, then there is a copy of G in K_n which is properly edge-coloured by c. This improves on a result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4), 409-433, 2003]. On the other hand, if c assigns each colour to at most n/51D^2 edges of K_n, then there is a copy of G in K_n such that each edge of G receives a different colour from c. This proves a conjecture of Frieze and Krivelevich [Electron. J. Comb. 15(1), R59, 2008]. Our proofs rely on a framework developed by Lu and Sz\'ekely [Electron. J. Comb. 14(1), R63, 2007] for applying the local lemma to random injections. In order to improve the constants in our results we use a version of the local lemma due to Bissacot, Fern\'andez, Procacci, and Scoppola [preprint, arXiv:0910.1824].Comment: 9 page

On Lennard-Jones type potentials and hard-core potentials with an attractive tail

We revisit an old tree graph formula, namely the Brydges-Federbush tree identity, and use it to get new bounds for the convergence radius of the Mayer series for gases of continuous particles interacting via non absolutely summable pair potentials with an attractive tail including Lennard-Jones type pair potentials