58 research outputs found
The generalized 3-connectivity of Cartesian product graphs
The generalized connectivity of a graph, which was introduced recently by
Chartrand et al., is a generalization of the concept of vertex connectivity.
Let be a nonempty set of vertices of , a collection
of trees in is said to be internally disjoint trees
connecting if and for
any pair of distinct integers , where . For an integer
with , the -connectivity of is the
greatest positive integer for which contains at least internally
disjoint trees connecting for any set of vertices of .
Obviously, is the connectivity of . Sabidussi showed
that for any two connected graphs
and . In this paper, we first study the 3-connectivity of the Cartesian
product of a graph and a tree , and show that if
, then ;
if , then .
Furthermore, for any two connected graphs and with
, if , then ; if , then
. Our result could be seen as
a generalization of Sabidussi's result. Moreover, all the bounds are sharp.Comment: 17 page
The generalized 3-edge-connectivity of lexicographic product graphs
The generalized -edge-connectivity of a graph is a
generalization of the concept of edge-connectivity. The lexicographic product
of two graphs and , denoted by , is an important graph
product. In this paper, we mainly study the generalized 3-edge-connectivity of
, and get upper and lower bounds of .
Moreover, all bounds are sharp.Comment: 14 page
The generalized 3-connectivity of Lexicographic product graphs
The generalized -connectivity of a graph , introduced by
Chartrand et al., is a natural and nice generalization of the concept of
(vertex-)connectivity. In this paper, we prove that for any two connected
graphs and , . We also give
upper bounds for and . Moreover, all
the bounds are sharp.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1103.609
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
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
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