12 research outputs found

    Extra Connectivity of Strong Product of Graphs

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    The gg-extraextra connectivityconnectivity κg(G)\kappa_{g}(G) of a connected graph GG is the minimum cardinality of a set of vertices, if it exists, whose deletion makes GG disconnected and leaves each remaining component with more than gg vertices, where gg is a non-negative integer. The strongstrong productproduct G1⊠G2G_1 \boxtimes G_2 of graphs G1G_1 and G2G_2 is the graph with vertex set V(G1⊠G2)=V(G1)×V(G2)V(G_1 \boxtimes G_2)=V(G_1)\times V(G_2), where two distinct vertices (x1,y1),(x2,y2)∈V(G1)×V(G2)(x_{1}, y_{1}),(x_{2}, y_{2}) \in V(G_1)\times V(G_2) are adjacent in G1⊠G2G_1 \boxtimes G_2 if and only if x1=x2x_{1}=x_{2} and y1y2∈E(G2)y_{1} y_{2} \in E(G_2) or y1=y2y_{1}=y_{2} and x1x2∈E(G1)x_{1} x_{2} \in E(G_1) or x1x2∈E(G1)x_{1} x_{2} \in E(G_1) and y1y2∈E(G2)y_{1} y_{2} \in E(G_2). In this paper, we give the g (≤3)g\ (\leq 3)-extraextra connectivityconnectivity of G1⊠G2G_1\boxtimes G_2, where GiG_i is a maximally connected ki (≥2)k_i\ (\geq 2)-regular graph for i=1,2i=1,2. As a byproduct, we get g (≤3)g\ (\leq 3)-extraextra conditional fault-diagnosability of G1⊠G2G_1\boxtimes G_2 under PMCPMC model

    The super-connectivity of Johnson graphs

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    For positive integers n,kn,k and tt, the uniform subset graph G(n,k,t)G(n, k, t) has all kk-subsets of {1,2,…,n}\{1,2,\ldots, n\} as vertices and two kk-subsets are joined by an edge if they intersect at exactly tt elements. The Johnson graph J(n,k)J(n,k) corresponds to G(n,k,k−1)G(n,k,k-1), that is, two vertices of J(n,k)J(n,k) are adjacent if the intersection of the corresponding kk-subsets has size k−1k-1. 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 J(n,k)J(n,k) for n≥k≥1n\geq k\geq 1

    The isoperimetric problem in Johnson graphs

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    It has been recently proved that the connectivity of distance regular graphs is the degree of the graph. We study the Johnson graphs J(n,m)J(n,m), which are not only distance regular but distance transitive, with the aim to analyze deeper connectivity properties in this class. The vertex kk-connectivity of a graph GG is the minimum number of vertices that have to be removed in order to separate the graph into two sets of at least kk vertices in each one. The isoperimetric function μG(k)\mu_G(k) of a graph GG is the minimum boundary among all subsets of vertices of fixed cardinality kk. We give the value of the isoperimetric function of the Johnson graph J(n,m)J(n,m) for values of kk of the form (tm){t\choose m}, 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 kk-connectivity of the Johnson graphs as well. We will see that the kk-connectivity grows very fast with kk, 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 J(n,3)J(n,3). . 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

    Master index of volumes 161–170

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    The generalized 4-connectivity of burnt pancake graphs

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    The generalized kk-connectivity of a graph GG, denoted by κk(G)\kappa_k(G), is the minimum number of internally edge disjoint SS-trees for any S⊆V(G)S\subseteq V(G) and ∣S∣=k|S|=k. The generalized kk-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. An nn-dimensional burnt pancake graph BPnBP_n is a Cayley graph which posses many desirable properties. In this paper, we try to evaluate the reliability of BPnBP_n by investigating its generalized 4-connectivity. By introducing the notation of inclusive tree and by studying structural properties of BPnBP_n, we show that κ4(BPn)=n−1\kappa_4(BP_n)=n-1 for n≥2n\ge 2, that is, for any four vertices in BPnBP_n, there exist (n−1n-1) internally edge disjoint trees connecting them in BPnBP_n

    Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010

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    Subject Index Volumes 1–200

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