The Fibonacci cube Γn is the subgraph of the hypercube 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. The eccentricity of a vertex u, denoted eG(u) is the greatest distance between u and any other vertex v in the graph G. For a given vertex u of Γn we characterize the vertices v such that d...

In this paper, a new class of hierarchically definable graphs are proposed and they are proper subgraphs of Hierarchic Cubic graphs. These graphs are based on the Fibonacci series by changing initial conditions. When the initial conditions are changed, then the structure of obtained graph will be changed. Thus, we obtained a series of hierarchically definable graphs. The obtained graphs have lo...

The Fibonacci cube of dimension n, denoted as $$\Gamma _n$$ , is the subgraph hypercube induced by vertices with no consecutive 1s. Lucas $$\Lambda cyclic version . irregularity a graph G sum $$|d(x)-d(y)|$$ over all edges $$\{x,y\}$$ G. In two recent papers based on recursive structure it proved that and times number _{n-1}$$ 2n _{n-4}$$ respectively. Using an interpretation in terms couples i...

a non-abelian finite group is called sequenceable if for some positive integer , is -generated ( ) and there exist integers such that every element of is a term of the -step generalized fibonacci sequence , , , . a remarkable application of this definition may be find on the study of random covers in the cryptography. the 2-step generalized sequences for the dihedral groups studied for their pe...

In this paper we introduce a family of infinite words that generalize the Fibonacci word and we study their combinatorial properties. We associate with this family of words a family of curves that are like the Fibonacci word fractal and reveal some fractal features. Finally, we describe an infinite family of polyominoes stems from the generalized Fibonacci words and we study some of their geome...

We present a new class of interconnection topologies called the Linear Recursive Networks (LRNs) and examine their possible applications in distributed systems. Each LRN is characterized by a recursive pattern of interconnection which can be specified by simple parameters. Basic properties such as node degree, diameter, and the performance of routing algorithms for all LRNs are then collectivel...

The Fibonacci (p, r)-cube is an interconnection topology, which includes a wide range of connection topologies as its special cases, such as the Fibonacci cube, the postal network, etc. Klavžar and Žigert [S. Klavžar, P. Žigert, Fibonacci cubes are the resonance graphs of fibonaccenes, Fibonacci Quart. 43 (2005) 269–276] proved that Fibonacci cubes are just the Z-transformation graphs (also cal...

The postal network is an interconnection network that possesses many desirable properties which are important in network design and applications. It includes hypercubes and Fibonacci cubes as its special cases. The postal network can also be considered as a flexible version of the hypercube which relaxes the restriction on the number of nodes and thus makes it possible to construct multicompute...

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we introduce a matricial toeplitz transform and prove that the toeplitz transform of a second order recurrence sequence is another second order recurrence sequence. we investigate the injectivity of this transform and show how this distinguishes the fibonacci sequence among other recurrence sequences. we then obtain new fibonacci identities as an application of our transform.