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For an ordered set of vertices and a vertex in a connected graph , the representation of with respect to is the -vector = (, , where represents the distance between the vertices and . The set is a resolving set for if distinct vertices of have distinct representations with respect to . A resolving set for containing a minimum number of vertices is a basis for . The dimension is the number of vertices in a basis for . A resolving set of is connected if the subgraph...
For an ordered -decomposition of a connected graph and an edge of , the -code of is the -tuple = (
We study two topological properties of the 5-ary -cube . Given two arbitrary distinct nodes and in , we prove that there exists an - path of every length ranging from to , where . Based on this result, we prove that is 5-edge-pancyclic by showing that every edge in lies on a cycle of every length ranging from to .
We study two topological properties of the 5-ary n-cube
. Given two arbitrary distinct nodes x and y in
, we prove that there exists an
x-y path of every length ranging from 2n to 5n - 1, where n ≥ 2. Based
on this result, we prove that is
5-edge-pancyclic by showing that every edge in lies on
a cycle of every length ranging from 5 to 5n.
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