31 research outputs found
Edges in Fibonacci cubes, Lucas cubes and complements
The Fibonacci cube of dimension n, denoted as , is the subgraph of
the hypercube induced by vertices with no consecutive 1's. The irregularity of
a graph G is the sum of |d(x)-d(y)| over all edges {x,y} of G. In two recent
paper based on the recursive structure of it is proved that the
irregularity of and are two times the number of edges
of and times the number of vertices of ,
respectively. Using an interpretation of the irregularity in terms of couples
of incident edges of a special kind (Figure 2) we give a bijective proof of
both results. For these two graphs we deduce also a constant time algorithm for
computing the imbalance of an edge. In the last section using the same approach
we determine the number of edges and the sequence of degrees of the cube
complement of
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
On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes
Abstract Let Γ n and Λ n be the n-dimensional Fibonacci cube and Lucas cube, respectively. The domination number γ of Fibonacci cubes and Lucas cubes is studied. In particular it is proved that γ(Λ n ) is bounded below by , where L n is the n-th Lucas number. The 2-packing number ρ of these cubes is also studied. It is proved that and the exact values of ρ(Γ n ) and ρ(Λ n ) are obtained for n ≤ 10. It is also shown that Aut(Γ n ) Z 2
Generalized Fibonacci cubes
AbstractGeneralized Fibonacci cube Qd(f) is introduced as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a substring. In this notation, the Fibonacci cube Γd is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and non-embeddable infinite series are given. The question is completely solved for strings f of length at most five and for strings consisting of at most three blocks. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial cases Qd(10) and Qd(01), the only generalized Fibonacci cubes that are median closed subgraphs of the corresponding hypercubes. For admissible strings f, the f-dimension of a graph is introduced. Several problems and conjectures are also listed
Algebraic Analysis of Vertex-Distinguishing Edge-Colorings
Vertex-distinguishing edge-colorings (vdec colorings) are a restriction of proper edge-colorings. These special colorings require that the sets of edge colors incident to every vertex be distinct. This is a relatively new field of study. We present a survey of known results concerning vdec colorings. We also define a new matrix which may be used to study vdec colorings, and examine its properties. We find several bounds on the eigenvalues of this matrix, as well as results concerning its determinant, and other properties. We finish by examining related topics and open problems