Abstract

Fault tolerance is especially important for interconnection networks, since the growing size of networks increases their vulnerability to component failures. A classical measure for the fault tolerance of a network in the case of vertex failures is its connectivities. A network based on a graph G = (X1?X2?...? Xk,I,E) is called a k-multisplit network, if its vertex set V can be partitioned into k+1 stable sets I,X1,X2,...,Xk such that X1 ?X2?...?Xk induces a complete k-partite graph and I is an independent set. In this note, we first show that: for any non-complete connected k-multisplit graph G = (X1?X2?...?Xk,I,E) with k ? 3 and |X1| ? |X2| ?...? |Xk|, each of the following holds (1) If |X1?X2?...?Xk-1| ? ?, then k(G) = ?(G). (2) If |X1?X2?...?Xk-1| < ?, then k(G)? |X1? X2?...? Xk-1|. (3) ?(G) = ?(G). (4) If |X1?X2?...?Xk-1| > ? with respect to |X1| ? 2 and ? ? 2, then G is super-k. (5) G is super-?. In addition, we present sufficient conditions for digraphs to be maximally edge-connected and super-edge connected in terms of the zeroth-order general Randic index of digraphs.

Highlights

  • Interconnection networks is an important research area for parallel and distributed computer systems

  • We present sufficient conditions for digraphs to be maximally edge-connected and super-edge connected in terms of the zeroth-order general Randic index of digraphs

  • We mainly explore sufficient conditions for k-multisplit graphs and digraphs to be super-edge-connected and maximally edge-connected, respectively

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Summary

INTRODUCTION

Interconnection networks is an important research area for parallel and distributed computer systems. Network reliability is one of the most significant factors. In designing the topology of an interconnection network. Fault tolerance is an important property of network performance. A classic measure of the fault tolerance of a network is the connectivity and edge-connectivity of the corresponding graph. The larger connectivity/edge-connectivity is, the more reliable the network is. A network can be conveniently modelled as a graph or digraph. We mainly explore sufficient conditions for k-multisplit graphs and digraphs to be super-edge-connected and maximally edge-connected, respectively. We begin with some notations and concepts for graphs and digraphs

Notations for graphs
Notations for digraphs
Motivation
MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED SPLIT GRAPHS
MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED DIGRAPHS
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