Linearly many faults in arrangement graphs

Abstract

The star graph proposed by Akers et al. (Proc Int ConfParallel Process, University Park, PA, 1987, pp. 393-400) has many advantages over the n-cube. However, it suffers from having large gaps in the possible number of vertices. The arrangement graph was proposed by Day and Tripathi (Inf Process Lett 42 (1992), 235-241) to address this issue. Since it is a generalization of the star graph, it retains many of the nice properties of the star graph. In fact, it also generalizes the alternating group graph (Jwo et al., Networks 23 (1993), 315-326). There are many different measures of structural integrity of interconnection networks. In this article, we prove results of the following type for the arrangement graph: If h(r , n, k ) vertices are deleted from the arrangement graph A n,k , the resulting graph will either be connected or have a large component and small components having at most r − 1 vertices in total. Our result is tight for r ≤ 3, and it is asymptotically tight for r ≥ 4. Moreover, we also determine the cyclic vertex-connectivity of the arrangement graph. large component and some small components with at most r −1 vertices in total. In section 2, we introduce the necessary definitions, in section 3, we examine what happens when the number of deleted vertices is at most three times the common degree, and in section 4 we examine the case of deleting linearly many vertices. DEFINITIONS AND PRELIMINARIESA graph G = (V , E) with vertex set V and edge set E is r-regular if the degree of every vertex of G is r. If W ⊂ V is a set of vertices of G, then the graph obtained by deleting the vertices of W from G will be denoted by G − W . A noncomplete graph G is r-connected if deleting any set of fewer than r vertices results in a connected graph. A complete graph with r + 1 vertices is k-connected for k ≤ r. An rregular graph is maximally connected if it is r-connected.

The Reliability of Subgraphs in the Arrangement Graph

“…Chiang and Chen [4] showed that is isomorphic to the -alternating group graph . Recent years have witnessed the continued interest in the research of arrangement graphs [8], [9], [13], [17], [19], [24], [25].…”

Cyclic Decomposition of $k$-Permutations and Eigenvalues of the Arrangement Graphs

“…Proof. The eigenvalues of A(4, 3) and A(5, 3) are determined by a computer; these are {−3 [1] , −2 [6] , −1 [3] , 0 [4] , 1 [3] , 2 [6] , 3 [1] } and {−3 [14] , −2 [5] , −1 [12] , 1 [14] , 2 [6] , 4 [8] , 6 [1] }, respectively, which agree with the assertion. (Note that letting n = 4 in the assertion, the sum of the multiplicities of −3 and n − 7 equals 1.)…”

“…Many properties of arrangement graphs have been studied by a number of authors, see, e.g. [4,5,6,7,8,19,20].Another family of graphs with the same nature as the arrangement graphs are the derangement graphs.The n-derangement graph is a graph whose vertices correspond to all the permutations of [n] where two permutations are adjacent if they differ in all n positions. It is known that the eigenvalues of the derangement graph are integers (see [1,11,15,17]).…”

The symmetry property of (n,k)‐arrangement graph

“…For example, the spectra of arrangement graphs were studied in [2,7], the faulty tolerances were investigated in [36,39], and the conditional fault tolerance and the fault diagnosability were given in [40,41]. For more works on the arrangement graphs, we refer the reader to [2,8–12,19,19,23,31,38] and the references therein. These investigations show that this class of graphs admit many nice properties, such as hierarchical structure, simple shortest path routing, high connectivity, and so on.…”