The k-metric dimension of a graph

As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of any pair of different vertices of $G$ are distinguished by at least $k$ elements of $S$, i.e., for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. A connected graph $G$ is $k$-metric dimensional if $k$ is the largest integer such that there exists a $k$-metric basis for $G$. We give a necessary and sufficient condition for a graph to be $k$-metric dimensional and we obtain several results on the $k$-metric dimension

Nuclear shape dependence of Gamow-Teller distributions in neutron-deficient Pb isotopes

We study Gamow-Teller strength distributions in the neutron-deficient even isotopes (184-194)Pb in a search for signatures of deformation. The microscopic formalism used is based on a deformed quasiparticle random phase approximation (QRPA) approach, which involves a self-consistent quasiparticle deformed Skyrme Hartree-Fock (HF) basis and residual spin-isospin forces in both the particle-hole and particle-particle channels. By analyzing the sensitivity of the Gamow-Teller strength distributions to the various ingredients in the formalism, we conclude that the beta-decay of these isotopes could be a useful tool to look for fingerprints of nuclear deformation.Comment: 20 pages, 11 figures. To be published in Physical Review

Papilla preservation periodontal surgery in periodontal reconstruction for deep combined intra-suprabony defects. Retrospective analysis of a registry-based cohort

Suprabony defects are the most prevalent defects and there is very little evidence on their treatment. This study aims to assess the effectiveness of papilla preservation periodontal surgery in the periodontal reconstruction of combined deep intra-suprab