11,871 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-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
Computing Reformulated First Zagreb Index of Some Chemical Graphs as an Application of Generalized Hierarchical Product of Graphs
The generalized hierarchical product of graphs was introduced by L.
Barri\'ere et al in 2009. In this paper, reformulated first Zagreb index of
generalized hierarchical product of two connected graphs and hence as a special
case cluster product of graphs are obtained. Finally using the derived results
the reformulated first Zagreb index of some chemically important graphs such as
square comb lattice, hexagonal chain, molecular graph of truncated cube, dimer
fullerene, zig-zag polyhex nanotube and dicentric dendrimers are computed.Comment: 12 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
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