246 research outputs found
The degree sequence of Fibonacci and Lucas cubes
AbstractThe Fibonacci cube Γn is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λn is obtained from Γn by removing vertices that start and end with 1. It is proved that the number of vertices of degree k in Γn and Λn is ∑i=0k(n−2ik−i)(i+1n−k−i+1) and ∑i=0k[2(i2i+k−n)(n−2i−1k−i)+(i−12i+k−n)(n−2ik−i)], respectively. Both results are obtained in two ways, since each of the approaches yields additional results on the degree sequences of these cubes. In particular, the number of vertices of high resp. low degree in Γn is expressed as a sum of few terms, and the generating functions are given from which the moments of the degree sequences of Γn and Λn are easily computed
Maximal hypercubes in Fibonacci and Lucas cubes
The Fibonacci cube is the subgraph of the hypercube induced by the
binary strings that contain no two consecutive 1's. The Lucas cube
is obtained from by removing vertices that start and end with 1. We
characterize maximal induced hypercubes in and and
deduce for any the number of maximal -dimensional hypercubes in
these graphs
On Disjoint hypercubes in Fibonacci cubes
The {\em Fibonacci cube} of dimension , denoted as , is the
subgraph of -cube induced by vertices with no consecutive 1's. We
study the maximum number of disjoint subgraphs in isomorphic to
, and denote this number by . We prove several recursive results
for , in particular we prove that . We also prove a closed formula in which is given in
terms of Fibonacci numbers, and finally we give the generating function for the
sequence
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