13 research outputs found
On Linkedness of Cartesian Product of Graphs
We study linkedness of Cartesian product of graphs and prove that the product
of an -linked and a -linked graphs is -linked if the graphs are
sufficiently large. Further bounds in terms of connectivity are shown. We
determine linkedness of product of paths and product of cycles
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
On average connectivity of the strong product of graphs
The average connectivity κ(G) of a graph G is the average, over all pairs of vertices, of the
maximum number of internally disjoint paths connecting these vertices. The connectivity
κ(G) can be seen as the minimum, over all pairs of vertices, of the maximum number
of internally disjoint paths connecting these vertices. The connectivity and the average
connectivity are upper bounded by the minimum degree δ(G) and the average degree d(G)
of G, respectively. In this paper the average connectivity of the strong product G1 G2 of two
connected graphs G1 and G2 is studied. A sharp lower bound for this parameter is obtained.
As a consequence, we prove that κ(G1 G2) = d(G1 G2) if κ(Gi) = d(Gi), i = 1, 2. Also
we deduce that κ(G1 G2) = δ(G1 G2) if κ(Gi) = δ(Gi), i = 1, 2.Ministerio de Educación y Ciencia MTM2011-28800-C02-02Generalitat de Cataluña 1298 SGR200
Polytopality and Cartesian products of graphs
We study the question of polytopality of graphs: when is a given graph the
graph of a polytope? We first review the known necessary conditions for a graph
to be polytopal, and we provide several families of graphs which satisfy all
these conditions, but which nonetheless are not graphs of polytopes. Our main
contribution concerns the polytopality of Cartesian products of non-polytopal
graphs. On the one hand, we show that products of simple polytopes are the only
simple polytopes whose graph is a product. On the other hand, we provide a
general method to construct (non-simple) polytopal products whose factors are
not polytopal.Comment: 21 pages, 10 figure
The Average Lower Connectivity of Graphs
For a vertex v of a graph G, the lower connectivity, denoted by sv(G), is the smallest number of vertices that contains v and those vertices whose deletion from G produces a disconnected or a trivial graph. The average lower connectivity denoted by κav(G) is the value (∑v∈VGsvG)/VG. It is shown that this parameter can be used to measure the vulnerability of networks. This paper contains results on bounds for the average lower connectivity and obtains the average lower connectivity of some graphs
On linkedness in the Cartesian product of graphs
We study linkedness of the Cartesian product of graphs and prove that the product of an a-linked and a b-linked graph is (a + b - 1)-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine linkedness of products of paths and products of cycles
The degree sequence on tensor and cartesian products of graphs and their omega index
The aim of this paper is to illustrate how degree sequences may successfully be used over some graph products. Moreover, by taking into account the degree sequence, we will expose some new distinguishing results on special graph products. We will first consider the degree sequences of tensor and cartesian products of graphs and will obtain the omega invariant of them. After that we will conclude that the set of graphs forms an abelian semigroup in the case of tensor product whereas this same set is actually an abelian monoid in the case of cartesian product. As a consequence of these two operations, we also give a result on distributive law which would be important for future studies
ON PATH-PAIRABILITY IN THE CARTESIAN PRODUCT OF GRAPHS
We study the inheritance of path-pairability in the Cartesian product of graphs and prove additive and multiplicative inheritance patterns of pathpairability, depending on the number of vertices in the Cartesian product. We present path-pairable graph families that improve the known upper bound on the minimal maximum degree of a path-pairable graph. Further results and open questions about path-pairability are also presented