A new kind of parameter for fault tolerance of graphs

Abstract

SummarySuppose that G = (V(G), E(G)) is a connected graph and D(G) is the diameter of G. For two distinct vertices a1, b1 ∈ V, κ(a1, b1), defined as local connectivity of a1 and b1, is the maximum value of independent paths from a1 to b1 in G. Similarly, λ(a1, b1) is local edge connectivity of a1 and b1. For some t ∈ [1, D(G)], ∀ a1, b1 ∈ V, a1 is distinct to b1, and distance of a1 and b1 is d(a1, b1) = t, if κ(a1, b1) or (λ(a1, b1)) = min {d(a1), d(b1)}, then G is called t‐distance optimally (edge) connected. For all integers 0 < k ≤ t, if G is k‐distance optimally connected, then we call that G is t‐distance local optimally connected. Similarly, we have the definition of t‐distance local optimally edge connected graphs. The t‐distance connectivity κd(G, t) of G is urn:x-wiley:15320626:media:cpe4787:cpe4787-math-0001 The distance connectivity of G is defined as κd(G) = (κd(G, 1), κd(G, 2), …, κd(G, D(G))). Then, λd(G, t) and λd(G) can be defined similarly. Using the number of (edge) independent vi‐vj paths to replace the number 1, we generalize the adjacency matrix to the (edge) path matrix. In this research, we characterize the distance connectivity of graphs with D(G) ≤ 2, Cartesian product of k‐regular graphs, and the regular graphs. We also determine the eigenvalues of path matrix of some graphs and give some problems about distance connectivity.

The 3-Good-Neighbor Connectivity of Modified Bubble-Sort Graphs

“…B [2,5] 5 is connected by Claim 1. By Proposition 9, each vertex in B 1 5 − F 1 has one outside neighbor; we have that B 1 5 − F 1 is connected to B [2,5]…”

“…If we delete vertex n from the cycle G(T), then it results in a path on (n − 1) vertices, and all edges of G(T) − n { } are the generating set of S n− 1 . We can partition MB n into n subgraphs MB 1 n , MB 2 n , . .…”

“…Because of a significant increase in the number of processors, the processor failure is inevitable. Hence, fault diagnosis and fault tolerance of interconnection networks have become increasingly important [1,2]. e modified bubble-sort graph has been studied by many researchers [3][4][5].…”

“…en B 1 5 − F 1 is disconnected. By Lemma 2, B 1 5 − F 1 has two components, one of which is an isolated vertex v 1 . Note that |F [2,5] [2,5] |, we have B 5 [V(C 1 ) ∪ V (B [2,5] 5 − F [2,5] )] is connected by Proposition 9.…”

“…] is connected by Proposition 5. us, MB [2,n] n − F [2,n] is connected. By Proposition 8, each vertex u ∈ V(MB 1 n − F 1 ) has at least one outside neighbor in MB [2,n] n − F [2,n] . us, MB n − F is connected.…”