4,012 research outputs found
On the super connectivity of Kronecker products of graphs
In this paper we present the super connectivity of Kronecker product of a
general graph and a complete graph.Comment: 8 page
The mincut graph of a graph
In this paper we introduce an intersection graph of a graph , with vertex
set the minimum edge-cuts of . We find the minimum cut-set graphs of some
well-known families of graphs and show that every graph is a minimum cut-set
graph, henceforth called a \emph{mincut graph}. Furthermore, we show that
non-isomorphic graphs can have isomorphic mincut graphs and ask the question
whether there are sufficient conditions for two graphs to have isomorphic
mincut graphs. We introduce the -intersection number of a graph , the
smallest number of elements we need in in order to have a family of subsets, such that for each subset. Finally we
investigate the effect of certain graph operations on the mincut graphs of some
families of graphs
Connectivity of Direct Products of Graphs
Let be the connectivity of and the direct product
of and . We prove that for any graphs and with ,
, which was conjectured
by Guji and Vumar.Comment: 5 pages, accepted by Ars Com
Connectivity Threshold for random subgraphs of the Hamming graph
We study the connectivity of random subgraphs of the -dimensional Hamming
graph , which is the Cartesian product of complete graphs on
vertices. We sample the random subgraph with an i.i.d.\ Bernoulli bond
percolation on with parameter . We identify the window of the
transition: when the probability that the graph is
connected goes to , while when it converges to
.
We also investigate the connectivity probability inside the critical window,
namely when .
We find that the threshold does not depend on , unlike the phase
transition of the giant connected component the Hamming graph (see [Bor et al,
2005]). Within the critical window, the connectivity probability does depend on
d. We determine how.Comment: 10 pages, no figure
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