16,662 research outputs found
Properties of stochastic Kronecker graphs
The stochastic Kronecker graph model introduced by Leskovec et al. is a
random graph with vertex set , where two vertices and
are connected with probability
independently of the presence or absence of any other edge, for fixed
parameters . They have shown empirically that the
degree sequence resembles a power law degree distribution. In this paper we
show that the stochastic Kronecker graph a.a.s. does not feature a power law
degree distribution for any parameters . In addition,
we analyze the number of subgraphs present in the stochastic Kronecker graph
and study the typical neighborhood of any given vertex.Comment: 37 pages, 2 figure
Offensive alliances in cubic graphs
An offensive alliance in a graph is a set of vertices
where for every vertex in its boundary it holds that the
majority of vertices in 's closed neighborhood are in . In the case of
strong offensive alliance, strict majority is required. An alliance is
called global if it affects every vertex in , that is, is a
dominating set of . The global offensive alliance number
(respectively, global strong offensive alliance number
) is the minimum cardinality of a global offensive
(respectively, global strong offensive) alliance in . If has
global independent offensive alliances, then the \emph{global independent
offensive alliance number} is the minimum cardinality among
all independent global offensive alliances of . In this paper we study
mathematical properties of the global (strong) alliance number of cubic graphs.
For instance, we show that for all connected cubic graph of order ,
where
denotes the line graph of . All the above bounds are tight
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