75 research outputs found
Gromov hyperbolicity in directed graphs
In this paper, we generalize the classical definition of Gromov hyperbolicity to the context
of directed graphs and we extend one of the main results of the theory: the equivalence of the Gromov
hyperbolicity and the geodesic stability. This theorem has potential applications to the development
of solutions for secure data transfer on the internetSupported in part by two grants from Ministerio de EconomÃa y Competititvidad, Agencia Estatal
de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spai
Alliance free and alliance cover sets
A \emph{defensive} (\emph{offensive}) -\emph{alliance} in
is a set such that every in (in the boundary of ) has
at least more neighbors in than it has in . A set
is \emph{defensive} (\emph{offensive}) -\emph{alliance free,}
if for all defensive (offensive) -alliance , ,
i.e., does not contain any defensive (offensive) -alliance as a subset.
A set is a \emph{defensive} (\emph{offensive})
-\emph{alliance cover}, if for all defensive (offensive) -alliance ,
, i.e., contains at least one vertex from each
defensive (offensive) -alliance of . In this paper we show several
mathematical properties of defensive (offensive) -alliance free sets and
defensive (offensive) -alliance cover sets, including tight bounds on the
cardinality of defensive (offensive) -alliance free (cover) sets
On the hyperbolicity constant in graphs
AbstractIf X is a geodesic metric space and x1,x2,x3∈X, a geodesic triangle T={x1,x2,x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if, for every geodesic triangle T in X, every side of T is contained in a δ-neighborhood of the union of the other two sides. We denote by δ(X) the sharpest hyperbolicity constant of X, i.e. δ(X)≔inf{δ≥0:X is δ-hyperbolic}. In this paper, we obtain several tight bounds for the hyperbolicity constant of a graph and precise values of this constant for some important families of graphs. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its number of edges, diameter and cycles. As a consequence of our results, we show that if G is any graph with m edges with lengths {lk}k=1m, then δ(G)≤∑k=1mlk/4, and δ(G)=∑k=1mlk/4 if and only if G is isomorphic to Cm. Moreover, we prove the inequality δ(G)≤12diamG for every graph, and we use this inequality in order to compute the precise value δ(G) for some common graphs
Oscillation results for a nonlinear fractional differential equation
In this paper, the authors work with a general formulation of the fractional derivative of Caputo type. They study oscillatory solutions of differential equations involving these general fractional derivatives. In particular, they extend the Kamenev-type oscillation criterion given by Baleanu et al. in 2015. In addition, we prove results on the existence and uniqueness of solutions for many of the equations considered. Also, they complete their study with some examples
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