In:
Concurrency and Computation: Practice and Experience, Wiley, Vol. 31, No. 10 ( 2019-05-25)
Abstract:
Suppose that G = ( V ( G ), E ( G )) is a connected graph and D ( G ) is the diameter of G . For two distinct vertices a 1 , b 1 ∈ V , κ ( a 1 , b 1 ) , defined as local connectivity of a 1 and b 1 , is the maximum value of independent paths from a 1 to b 1 in G . Similarly, λ ( a 1 , b 1 ) is local edge connectivity of a 1 and b 1 . For some t ∈ [1, D ( G )], ∀ a 1 , b 1 ∈ V , a 1 is distinct to b 1 , and distance of a 1 and b 1 is d ( a 1 , b 1 ) = t , if κ ( a 1 , b 1 ) or ( λ ( a 1 , b 1 )) = min { d ( a 1 ), d ( b 1 )} , 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 κ d ( G , t ) of G is urn:x-wiley:15320626:media:cpe4787:cpe4787-math-0001 The distance connectivity of G is defined as κ d ( G ) = ( κ d ( G , 1), κ d ( G , 2), …, κ d ( G , D ( G ))) . Then, λ d ( G , t ) and λ d ( G ) can be defined similarly. Using the number of (edge) independent v i ‐ v j 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.
Type of Medium:
Online Resource
ISSN:
1532-0626
,
1532-0634
Language:
English
Publisher:
Wiley
Publication Date:
2019
detail.hit.zdb_id:
2052606-4
SSG:
11
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