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

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]

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

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

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 Stellar Transformation: From Interconnection Networks to Datacenter Networks

The first dual-port server-centric datacenter network, FiConn, was introduced in 2009 and there are several others now in existence; however, the pool of topologies to choose from remains small. We propose a new generic construction, the stellar transformation, that dramatically increases the size of this pool by facilitating the transformation of well-studied topologies from interconnection networks, along with their networking properties and routing algorithms, into viable dual-port server-centric datacenter network topologies. We demonstrate that under our transformation, numerous interconnection networks yield datacenter network topologies with potentially good, and easily computable, baseline properties. We instantiate our construction so as to apply it to generalized hypercubes and obtain the datacenter networks GQ*. Our construction automatically yields routing algorithms for GQ* and we empirically compare GQ* (and its routing algorithms) with the established datacenter networks FiConn and DPillar (and their routing algorithms); this comparison is with respect to network throughput, latency, load balancing, fault-tolerance, and cost to build, and is with regard to all-to-all, many all-to-all, butterfly, and random traffic patterns. We find that GQ* outperforms both FiConn and DPillar (sometimes significantly so) and that there is substantial scope for our stellar transformation to yield new dual-port server-centric datacenter networks that are a considerable improvement on existing ones.Comment: Submitted to a journa

Embedded connectivity of recursive networks

Let $G_n$ be an $n$-dimensional recursive network. The $h$-embedded connectivity $\zeta_h(G_n)$ (resp. edge-connectivity $\eta_h(G_n)$) of $G_n$ is the minimum number of vertices (resp. edges) whose removal results in disconnected and each vertex is contained in an $h$-dimensional subnetwork $G_h$. This paper determines $\zeta_h$ and $\eta_h$ for the hypercube $Q_n$ and the star graph $S_n$, and $\eta_3$ for the bubble-sort network $B_n$

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

A New Fault-Tolerant M-network and its Analysis

This paper introduces a new class of efficient inter connection networks called as M-graphs for large multi-processor systems.The concept of M-matrix and M-graph is an extension of Mn-matrices and Mn-graphs.We analyze these M-graphs regarding their suitability for large multi-processor systems. An(p,N) M-graph consists of N nodes, where p is the degree of each node.The topology is found to be having many attractive features prominent among them is the capability of maximal fault-tolerance, high density and constant diameter.It is found that these combinatorial structures exibit some properties like symmetry,and an inter-relation with the nodes, and degree of the concerned graph, which can be utilized for the purposes of inter connected networks.But many of the properties of these mathematical and graphical structures still remained unexplored and the present aim of the paper is to study and analyze some of the properties of these M-graphs and explore their application in networks and multi-processor systems

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