The objective of this study was to develop a new optimal parallel algorithm for matrix multiplication which could run on a FIBONACCI HYPERCUBE structure. Most of the popular algorithms for parallel matrix multiplication can not run on FIBONACCI HYPERCUBE structure, therefore giving a method that can be run on all structures especially FIBONACCI HYPERCUBE structure is necessary for parallel matrix multiplication. For creating this method, a new model for matrix multiplication with an algorithm for data distribution on FIBONACCI HYPERCUBE structure was provided. Other than this, another optimized algorithm was designed on Mesh structure. By running the algorithms on a simulative parallel system and giving the results in graphical mode, it has been found that these two algorithms have optimized value in parallel matrix multiplication and they are more efficient than the previous algorithms.

In this paper, we give two new optimal parallel algorithms for matrix multiplication which are designed to run on a FIBONACCI HYPERCUBE structure. At first, we present a broadcast algorithm on FIBONACCI HYPERCUBE with O (n) time complexity. We use this broadcast algorithm and algorithms presented by Gunnels, Lin, Morrow and Geijn on a mesh structure, in order to present two optimal parallel matrix multiplication algorithms on a FIBONACCI HYPERCUBE structure. We show these algorithms are cost-optimal. The performance of the algorithm has been tested on a simulative parallel system.

Body centered structures are used as seeds for a variety of structures of rank 3 and higher. Rhombellane structures are introduced and their design and topological properties are detailed.

In this paper, a new interconnection network structure called hierarchical extended FIBONACCI cubes interconnection networks HEFC1(n) is proposed, and its properties are evaluated. A set of the HEFC1(n)s constructed by the proposed method is a two level conventional hierarchical network. For each n (dimension of extended FIBONACCI cube-EFC1(n)), a HEFC1(n)   interconnection network can be constructed. All EFC1(n) for n>4 are recursively constructible, hence HEFC1(n)s for n>4 are also recursively constructible with some extra EFC1(n-1)s and EFC1(n-2)s. Furthermore, since a HEFC1(n) has a hierarchically structured character and the feature of uniformity, a wide variety of inter-cluster connections are possible. The comparisons of HEFC1(n)s with some of the traditional cubic and hierarchical cubic networks are presented in this study. The routing in HEFC1(n+2)s is as easy as routing in HCN(n,n) and the scalability of HEFC1(n+2) is better than the scalability of HCN(n,n) and H(2n). H(2n) and HCN(n,n) are symmetric interconnection networks, however, HEFC1(n+2) is not a symmetric interconnection network and HEFC1(n+2)s have a recurrent structure as H(2n) and HCN(n,n). The cost of HEFC1(n+2) is better than the costs of HCN(n,n) and H(2n).The edge connectivity of HEFC1(n+2) is worse than the edge connectivity of H(2n) and HCN(n,n). This makes the VLSI design of HEFC1(n+2) simpler than the VLSI designs of H(2n) and HCN(n,n).

We study two foremost Mahonian statistics, the major index and the inversion number for a class of binary words called restricted FIBONACCI words. The language of restricted FIBONACCI words satisfies recurrences which allow for the calculation of the generating functions in two different ways. These yield identities involving the $q$-binomial coefficients and provide non-standard $q$-analogues of the FIBONACCI numbers. The major index generating function for restricted FIBONACCI words turns out to be a $q$-power multiple of the inversion generating function.

The folded HYPERCUBE $FQ_n$ is the Cayley graph Cay$(\mathbb{Z}_2^n,S)$, where $S=\{e_1,e_2,\dots,e_n\}\cup \{u=e_1+e_2+\dots+e_n\}$, and $e_i = (0,\dots, 0, 1, 0,$ $\dots, 0)$, with 1 at the $i$th position, $1\leq i \leq n$. In this paper, we show that the folded HYPERCUBE $FQ_n$ is a distance-transitive graph. Then, we study some properties of this graph. In particular, we show that if $n\geq 4$ is an even integer, then the folded HYPERCUBE $FQ_n$ is an $automorphic$ graph, that is, $FQ_n$ is a distance-transitive primitive graph which is not a complete or a line graph.

The Wiener index and the Mostar index quantify two distance related properties of connected graphs: the Wiener index is the sum of the distances over all pairs of vertices and the Mostar index is a measure of how far the graph is from being distance-balanced. These two measures have been considered for a number of interesting families of graphs. In this paper, we determine the Wiener index and the Mostar index of alternate Lucas cubes. Alternate Lucas cubes form a family of interconnection networks whose recursive construction mimics the construction of the well-known FIBONACCI cubes.

Wireless sensor networks are composed of very small devices, called sensor nodes, for numerous applications in the environment. In adversarial environments, the security becomes a crucial issue in wireless sensor networks (WSNs). There are various security services in WSNs such as key management, authentication, and pairwise key establishment. Due to some limitations on sensor nodes, the previous key establishment techniques are unsuitable for WSNs. To overcome these problems, researchers propose several key pre-distribution schemes. Our proposed approach uses a combinatorial framework in the HYPERCUBE-based (HB) scheme to pre-distribute keys to each sensor node. By this way, the number of common keys between two nodes in a wireless communication range increases. Therefore, the level of security in terms of resilience against node capture attack and the probability of re-establishing an indirect key will be improved.

The aim of this paper is to introduce the cyclic-FIBONACCI hybrid sequence and give some properties. By taking into account the cyclic-FIBONACCI hybrid sequence modulo m, the method will be given to determine the period lengths of this sequence according to the different m values. In the final part of this paper, we study the cyclic-FIBONACCI hybrid sequence in groups and then we calculate the cyclic-FIBONACCI hybrid lengths of polyhedral groups (2, 2, 2), (2, n, 2) and (n, 2, 2) as applications of the results produced.