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 $B_n$

Super edge-connectivity and matching preclusion of data center networks

Edge-connectivity is a classic measure for reliability of a network in the presence of edge failures. $k$-restricted edge-connectivity is one of the refined indicators for fault tolerance of large networks. Matching preclusion and conditional matching preclusion are two important measures for the robustness of networks in edge fault scenario. In this paper, we show that the DCell network $D_{k,n}$ is super-$\lambda$ for $k\geq2$ and $n\geq2$, super-$\lambda_2$ for $k\geq3$ and $n\geq2$, or $k=2$ and $n=2$, and super-$\lambda_3$ for $k\geq4$ and $n\geq3$. Moreover, as an application of $k$-restricted edge-connectivity, we study the matching preclusion number and conditional matching preclusion number, and characterize the corresponding optimal solutions of $D_{k,n}$. In particular, we have shown that $D_{1,n}$ is isomorphic to the $(n,k)$-star graph $S_{n+1,2}$ for $n\geq2$.Comment: 20 pages, 1 figur

On restricted edge-connectivity of replacement product graphs

This paper considers the edge-connectivity and restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, the authors construct a Cayley graph with degree $d$ whose restricted edge-connectivity is equal to $d+s$ for given odd integer $d$ and integer $s$ with $d \geqslant 5$ and $1\leqslant s\leqslant d-3$, which answers a problem proposed ten years ago.Comment: 18 pages, 5 figures, 36 conference

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 restricted hh-connectivity of balanced hypercubes

The restricted $h$-connectivity of a graph $G$, denoted by $\kappa^h(G)$, is defined as the minimum cardinality of a set of vertices $F$ in $G$, if exists, whose removal disconnects $G$ and the minimum degree of each component of $G-F$ is at least $h$. In this paper, we study the restricted $h$-connectivity of the balanced hypercube $BH_n$ and determine that $\kappa^1(BH_n)=\kappa^2(BH_n)=4n-4$ for $n\geq2$. We also obtain a sharp upper bound of $\kappa^3(BH_n)$ and $\kappa^4(BH_n)$ of $n$-dimension balanced hypercube for $n\geq3$ ($n\neq4$). In particular, we show that $\kappa^3(BH_3)=\kappa^4(BH_3)=12$

Paired many-to-many 2-disjoint path cover of balanced hypercubes with faulty edges

As a variant of the well-known hypercube, the balanced hypercube $BH_n$ was proposed as a novel interconnection network topology for parallel computing. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2\}$ and $T=\{t_1,t_2\}$ are any two sets of two vertices in different partite sets of $BH_n$ ($n\geq1$). It has been proved that there exist two vertex-disjoint $s_1,t_1$-path and $s_2,t_2$-path of $BH_n$ covering all vertices of it. In this paper, we prove that there always exist two vertex-disjoint $s_1,t_1$-path and $s_2,t_2$-path covering all vertices of $BH_n$ with at most $2n-3$ faulty edges. The upper bound $2n-3$ of edge faults tolerated is optimal.Comment: 30 pages, 9 figure

Structure and substructure connectivity of balanced hypercubes

The connectivity of a network directly signifies its reliability and fault-tolerance. Structure and substructure connectivity are two novel generalizations of the connectivity. Let $H$ be a subgraph of a connected graph $G$. The structure connectivity (resp. substructure connectivity) of $G$, denoted by $\kappa(G;H)$ (resp. $\kappa^s(G;H)$), is defined to be the minimum cardinality of a set $F$ of connected subgraphs in $G$, if exists, whose removal disconnects $G$ and each element of $F$ is isomorphic to $H$ (resp. a subgraph of $H$). In this paper, we shall establish both $\kappa(BH_n;H)$ and $\kappa^s(BH_n;H)$ of the balanced hypercube $BH_n$ for $H\in\{K_1,K_{1,1},K_{1,2},K_{1,3},C_4\}$.Comment: arXiv admin note: text overlap with arXiv:1805.0846

Positive Grassmannian and polyhedral subdivisions

The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial structures appeared in many other areas of mathematics and physics, e.g., in the study of cluster algebras, scattering amplitudes, and solitons. We discuss new ways to think about these structures. In particular, we identify plabic graphs and more general Grassmannian graphs with polyhedral subdivisions induced by 2-dimensional projections of hypersimplices. This implies a close relationship between the positive Grassmannian and the theory of fiber polytopes and the generalized Baues problem. This suggests natural extensions of objects related to the positive Grassmannian.Comment: 25 page

Vulnerability of super edge-connected graphs

A subset $F$ of edges in a connected graph $G$ is a $h$-extra edge-cut if $G-F$ is disconnected and every component has more than $h$ vertices. The $h$-extra edge-connectivity \la^{(h)}(G) of $G$ is defined as the minimum cardinality over all $h$-extra edge-cuts of $G$. A graph $G$, if \la^{(h)}(G) exists, is super-\la^{(h)} if every minimum $h$-extra edge-cut of $G$ isolates at least one connected subgraph of order $h+1$. The persistence $\rho^{(h)}(G)$ of a super-\la^{(h)} graph $G$ is the maximum integer $m$ for which $G-F$ is still super-\la^{(h)} for any set $F\subseteq E(G)$ with $|F|\leqslant m$. Hong {\it et al.} [Discrete Appl. Math. 160 (2012), 579-587] showed that \min\{\la^{(1)}(G)-\delta(G)-1,\delta(G)-1\}\leqslant \rho^{(0)}(G)\leqslant \delta(G)-1, where $\delta(G)$ is the minimum vertex-degree of $G$. This paper shows that \min\{\la^{(2)}(G)-\xi(G)-1,\delta(G)-1\}\leqslant \rho^{(1)}(G)\leqslant \delta(G)-1, where $\xi(G)$ is the minimum edge-degree of $G$. In particular, for a $k$-regular super-\la' graph $G$, $\rho^{(1)}(G)=k-1$ if \la^{(2)}(G) does not exist or $G$ is super-\la^{(2)} and triangle-free, from which the exact values of $\rho^{(1)}(G)$ are determined for some well-known networks

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