A kind of conditional connectivity of transposition networks generated by kk-trees

For a graph $G = (V, E)$, a subset $F\subset V(G)$ is called an $R_k$-vertex-cut of $G$ if $G -F$ is disconnected and each vertex $u \in V(G)- F$ has at least $k$ neighbors in $G -F$. The $R_k$-vertex-connectivity of $G$, denoted by $\kappa^k(G)$, is the cardinality of the minimum $R_k$-vertex-cut of $G$, which is a refined measure for the fault tolerance of network $G$. In this paper, we study $\kappa^2$ for Cayley graphs generated by $k$-trees. Let $Sym(n)$ be the symmetric group on $\{1, 2, \cdots ,n\}$ and $\mathcal{T}$ be a set of transpositions of $Sym(n)$. Let $G(\mathcal{T})$ be the graph on $n$ vertices $\{1, 2, . . . ,n\}$ such that there is an edge $ij$ in $G(\mathcal{T})$ if and only if the transposition $ij\in \mathcal{T}$. The graph $G(\mathcal{T})$ is called the transposition generating graph of $\mathcal{T}$. We denote by $Cay(Sym(n),\mathcal{T})$ the Cayley graph generated by $G(\mathcal{T})$. The Cayley graph $Cay(Sym(n),\mathcal{T})$ is denoted by $T_kG_n$ if $G(\mathcal{T})$ is a $k$-tree. We determine $\kappa^2(T_kG_n)$ in this work. The trees are $1$-trees, and the complete graph on $n$ vertices is a $n-1$-tree. Thus, in this sense, this work is a generalization of the such results on Cayley graphs generated by transposition generating trees and the complete-transposition graphs.Comment: 11pages,2figure

Relationship between Conditional Diagnosability and 2-extra Connectivity of Symmetric Graphs

The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault-tolerance in a multiprocessor system. The conditional diagnosability $t_c(G)$ of $G$ is the maximum number $t$ for which $G$ is conditionally $t$-diagnosable under the comparison model, while the 2-extra connectivity $\kappa_2(G)$ of a graph $G$ is the minimum number $k$ for which there is a vertex-cut $F$ with $|F|=k$ such that every component of $G-F$ has at least $3$ vertices. A quite natural problem is what is the relationship between the maximum and the minimum problem? This paper partially answer this problem by proving $t_c(G)=\kappa_2(G)$ for a regular graph $G$ with some acceptable conditions. As applications, the conditional diagnosability and the 2-extra connectivity are determined for some well-known classes of vertex-transitive graphs, including, star graphs, $(n,k)$-star graphs, alternating group networks, $(n,k)$-arrangement graphs, alternating group graphs, Cayley graphs obtained from transposition generating trees, bubble-sort graphs, $k$-ary $n$-cube networks and dual-cubes. Furthermore, many known results about these networks are obtained directly

Generalized Measures of Fault Tolerance in (n,k)-star Graphs

This paper considers a kind of generalized measure $\kappa_s^{(h)}$ of fault tolerance in the $(n,k)$-star graph $S_{n,k}$ and determines $\kappa_s^{(h)}(S_{n,k})=n+h(k-2)-1$ for $2 \leqslant k \leqslant n-1$ and $0\leqslant h \leqslant n-k$, which implies that at least $n+h(k-2)-1$ vertices of $S_{n,k}$ have to remove to get a disconnected graph that contains no vertices of degree less than $h$. This result contains some known results such as Yang et al. [Information Processing Letters, 110 (2010), 1007-1011].Comment: 7 page, 9 reference

Generalized Measures of Fault Tolerance in Exchanged Hypercubes

The exchanged hypercube $EH(s,t)$, proposed by Loh {\it et al.} [The exchanged hypercube, IEEE Transactions on Parallel and Distributed Systems 16 (9) (2005) 866-874], is obtained by removing edges from a hypercube $Q_{s+t+1}$. This paper considers a kind of generalized measures $\kappa^{(h)}$ and $\lambda^{(h)}$ of fault tolerance in $EH(s,t)$ with $1\leqslant s\leqslant t$ and determines $\kappa^{(h)}(EH(s,t))=\lambda^{(h)}(EH(s,t))= 2^h(s+1-h)$ for any $h$ with $0\leqslant h\leqslant s$. The results show that at least $2^h(s+1-h)$ vertices (resp. $2^h(s+1-h)$ edges) of $EH(s,t)$ have to be removed to get a disconnected graph that contains no vertices of degree less than $h$, and generalizes some known results

Generalized Connectivity of Star Graphs

This paper shows that, for any integers $n$ and $k$ with $0\leqslant k \leqslant n-2$, at least $(k+1)!(n-k-1)$ vertices or edges have to be removed from an $n$-dimensional star graph to make it disconnected and no vertices of degree less than $k$. The result gives an affirmative answer to the conjecture proposed by Wan and Zhang [Applied Mathematics Letters, 22 (2009), 264-267]

From Graph Isoperimetric Inequality to Network Connectivity -- A New Approach

We present a new, novel approach to obtaining a network's connectivity. More specifically, we show that there exists a relationship between a network's graph isoperimetric properties and its conditional connectivity. A network's connectivity is the minimum number of nodes, whose removal will cause the network disconnected. It is a basic and important measure for the network's reliability, hence its overall robustness. Several conditional connectivities have been proposed in the past for the purpose of accurately reflecting various realistic network situations, with extra connectivity being one such conditional connectivity. In this paper, we will use isoperimetric properties of the hypercube network to obtain its extra connectivity. The result of the paper for the first time establishes a relationship between the age-old isoperimetric problem and network connectivity.Comment: 17 pages, 0 figure

The 4-Component Connectivity of Alternating Group Networks

The $\ell$-component connectivity (or $\ell$-connectivity for short) of a graph $G$, denoted by $\kappa_\ell(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This generalization is a natural extension of the classical connectivity defined in term of minimum vertex-cut. As an application, the $\ell$-connectivity can be used to assess the vulnerability of a graph corresponding to the underlying topology of an interconnection network, and thus is an important issue for reliability and fault tolerance of the network. So far, only a little knowledge of results have been known on $\ell$-connectivity for particular classes of graphs and small $\ell$'s. In a previous work, we studied the $\ell$-connectivity on $n$-dimensional alternating group networks $AN_n$ and obtained the result $\kappa_3(AN_n)=2n-3$ for $n\geqslant 4$. In this sequel, we continue the work and show that $\kappa_4(AN_n)=3n-6$ for $n\geqslant 4$

The gg-good neighbour diagnosability of hierarchical cubic networks

Let $G=(V, E)$ be a connected graph, a subset $S\subseteq V(G)$ is called an $R^{g}$-vertex-cut of $G$ if $G-F$ is disconnected and any vertex in $G-F$ has at least $g$ neighbours in $G-F$. The $R^{g}$-vertex-connectivity is the size of the minimum $R^{g}$-vertex-cut and denoted by $\kappa^{g}(G)$. Many large-scale multiprocessor or multi-computer systems take interconnection networks as underlying topologies. Fault diagnosis is especially important to identify fault tolerability of such systems. The $g$-good-neighbor diagnosability such that every fault-free node has at least $g$ fault-free neighbors is a novel measure of diagnosability. In this paper, we show that the $g$-good-neighbor diagnosability of the hierarchical cubic networks $HCN_{n}$ under the PMC model for $1\leq g\leq n-1$ and the $MM^{*}$ model for $1\leq g\leq n-1$ is $2^{g}(n+2-g)-1$, respectively

On restricted edge-connectivity of half-transitive multigraphs

Let $G=(V,E)$ be a multigraph (it has multiple edges, but no loops). The edge connectivity, denoted by $\lambda(G)$, is the cardinality of a minimum edge-cut of $G$. We call $G$ maximally edge-connected if $\lambda(G)=\delta(G)$, and $G$ super edge-connected if every minimum edge-cut is a set of edges incident with some vertex. The restricted edge-connectivity $\lambda'(G)$ of $G$ is the minimum number of edges whose removal disconnects $G$ into non-trivial components. If $\lambda'(G)$ achieves the upper bound of restricted edge-connectivity, then $G$ is said to be $\lambda'$-optimal. A bipartite multigraph is said to be half-transitive if its automorphism group is transitive on the sets of its bipartition. In this paper, we will characterize maximally edge-connected half-transitive multigraphs, super edge-connected half-transitive multigraphs, and $\lambda'$-optimal half-transitive multigraphs

Cayley graphs and symmetric interconnection networks

These lecture notes are on automorphism groups of Cayley graphs and their applications to optimal fault-tolerance of some interconnection networks. We first give an introduction to automorphisms of graphs and an introduction to Cayley graphs. We then discuss automorphism groups of Cayley graphs. We prove that the vertex-connectivity of edge-transitive graphs is maximum possible. We investigate the automorphism group and vertex-connectivity of some families of Cayley graphs that have been considered for interconnection networks; we focus on the hypercubes, folded hypercubes, Cayley graphs generated by transpositions, and Cayley graphs from linear codes. New questions and open problems are also discussed.Comment: A. Ganesan, "Cayley graphs and symmetric interconnection networks," Proceedings of the Pre-Conference Workshop on Algebraic and Applied Combinatorics (PCWAAC 2016), 31st Annual Conference of the Ramanujan Mathematical Society, pp. 118--170, Trichy, Tamilnadu, India, June 201