12 research outputs found
Extra Connectivity of Strong Product of Graphs
The - of a connected graph is
the minimum cardinality of a set of vertices, if it exists, whose deletion
makes disconnected and leaves each remaining component with more than
vertices, where is a non-negative integer. The of graphs and is the graph with vertex set , where two distinct vertices are adjacent in if and only if and or
and or and . In this paper, we give the - of
, where is a maximally connected -regular graph for . As a byproduct, we get -
conditional fault-diagnosability of under model
The super-connectivity of Johnson graphs
For positive integers and , the uniform subset graph
has all -subsets of as vertices and two -subsets are
joined by an edge if they intersect at exactly elements. The Johnson graph
corresponds to , that is, two vertices of are
adjacent if the intersection of the corresponding -subsets has size . A
super vertex-cut of a connected graph is a set of vertices whose removal
disconnects the graph without isolating a vertex and the super-connectivity is
the size of a minimum super vertex-cut. In this work, we fully determine the
super-connectivity of the family of Johnson graphs for
The isoperimetric problem in Johnson graphs
It has been recently proved that the connectivity of distance regular graphs is the degree of the graph. We study the Johnson graphs , which are not only distance regular but distance transitive, with the aim to analyze deeper connectivity properties in this class.
The vertex -connectivity of a graph is the minimum number of vertices that have to be removed in order to separate the graph into two sets of at least vertices in each one. The isoperimetric function of a graph is the minimum boundary among all subsets of vertices of fixed cardinality . We give the value of the isoperimetric function of the Johnson graph for values of of the form , and provide lower and upper bounds for this function for a wide range of its parameter. The computation of the isoperimetric function is used to study the -connectivity of the Johnson graphs as well. We will see that the -connectivity grows very fast with , providing much sensible information about the robustness of these graphs than just the ordinary connectivity.
In order to study the isoperimetric function of Johnson graphs we use combinatorial and spectral tools. The combinatorial tools are based on compression techniques, which allow us to transform sets of vertices without increasing their boundary. In the compression process we will show that sets of vertices that induce Johnson subgraphs are optimal with respect to the isoperimetric problem. Upper bounds are obtained by displaying nested families of sets which interpolate optimal ones. The spectral tools are used to obtain lower bounds for the isoperimetric function. These tools allow us to display completely the isoperimetric function for Johnson graphs . . Distance regular graphs form a structured class of graphs which include well-known families, as the n-cubes. Isoperimetric inequalities are well understood for the cubes, but for the general class of distance regular class it has only been proved that the connectivity of these graphs equals the degree. As another test case, the project suggests to study the family of so-called Johnson graphs, which are not only distance regular but also distance transitive. Combinatorial and spectral techniques to analyze the problem are available
The generalized 4-connectivity of burnt pancake graphs
The generalized -connectivity of a graph , denoted by , is
the minimum number of internally edge disjoint -trees for any and . The generalized -connectivity is a natural extension of
the classical connectivity and plays a key role in applications related to the
modern interconnection networks. An -dimensional burnt pancake graph
is a Cayley graph which posses many desirable properties. In this paper, we try
to evaluate the reliability of by investigating its generalized
4-connectivity. By introducing the notation of inclusive tree and by studying
structural properties of , we show that for , that is, for any four vertices in , there exist () internally
edge disjoint trees connecting them in