On the Padovan Codes and the Padovan Cubes

Abstract

We present a new interconnection topology called the Padovan cube. Despite their asymmetric and relatively sparse interconnections, the Padovan cubes are shown to possess attractive recurrent structures. Since they can be embedded in a subgraph of the Boolean cube and can have a Fibonacci cube as a subgraph, and since they are also a supergraph of other structures, it is possible that the Padovan cubes can be useful in fault-tolerant computing. For a graph with n vertices, we characterize the Padovan cubes. We also include the number of edges, decompositions, and embeddings, as well as the diameter of the Padovan cubes.

On the Cube Polynomials of Padovan and Lucas–Padovan Cubes

“…First, we need definitions of the Padovan cubes and the Ludovan cubes. In [10,11], the authors gave the definitions of the Padovan cube and the Ludovan cube by using the odd-Padovan sequence {a n } and the odd-Ludovan sequence {l n }, respectively.…”

“…In [10], the authors gave the following theorem for a characterization of the Padovan cubes P n . Theorem 3.…”

“…In (10), we have, from (9), ∑ n≥4 c 1 (P n+1 )y n = y ∑ n≥3 c 1 (P n+1 )y n + y 2 ∑ n≥2 c 1 (P n+1 )y n + y 4 g(c 1 (P n+1 ), y)…”

“…In [10,11], the authors introduced a new interconnection called the Padovan cube and Lucas-Padovan cube by using the Padovan sequence and Lucas-Padovan sequence, respectively. They gave a characterization of the Padovan cube and Lucas-Padovan cube, respectively.…”

“…Lee and Kim [10] introduced the Padovan cube by using the odd-Padovan sequence. The odd-Padovan sequence {a n } is the sequence of integers defined by a n = P 2n+1 for n ≥ 1.…”