In this paper, we introduced $m$-extension dual Complex Fibonacci $p$-numbers. We established the properties of $m$-extension dual Complex Fibonacci $p$-numbers. They are connected to Complex Fibonacci numbers, Complex Fibonacci $p$-numbers and dual Complex Fibonacci $p$-numbers.

The energyE (G) of a graph G is the sum of the absolute values of the eigenvalues ofG. In this article we consider the problem whether generalized Fibonacci constants jn (n³2) can be the energy of graphs. We show that jn cannot be the energy of graphs. Also we prove that all natural powers of j2n cannot be the energy of a matroid.

Quantum calculus, which arises in the mathematical fields of combinatorics and special functions as well as in a number of areas, involving the study of fractals and multi−, fractal measures, and expressions for the entropy of chaotic dynamical systems, has attracted the attention of many researchers in recent years. In this paper, by virtue of some useful notations from q−, calculus, we define the q−, Fibonacci dual biComplex numbers and q−, Lucas dual biComplex numbers with a different perspective. Afterwards, we give the Binet formulas, binomial sums, exponential generating functions, Catalan identities, Cassini identities, d’, Ocagne identities and some algebraic properties for the q−, Fibonacci dual biComplex numbers and q−, Lucas dual biComplex numbers.

In the present paper, we aim to generalize the notion of Complex-type Fibonacci sequences to Complex-type cyclic Fibonacci sequences. Firstly, we define the Complex-type cyclic-Fibonacci sequence and then we give miscellaneous  properties of this sequence by using the matrix method. Also, we study the Complex-type cyclic-Fibonacci sequence modulo $m$. In addition, we describe the Complex-type cyclic-Fibonacci sequence in a $2$-generator group and investigate that in finite groups in details. Then, as our last result, we obtain the lengths of the periods of the Complex-type cyclic-Fibonacci sequences in dihedral groups $D_{2}$, $D_{3}$, $D_{4}$, $D_{5}$, $D_{6}$ and $D_{8}$ with respect to their generating sets.

In this paper, a class of new polynomials based on Fibonacci sequence using Newton interpolation is introduced. This target is performed once using Newton forward-divided-difference formula and another more using Newton backward-divided-difference formula. Some interesting results are obtained for forward and backward differences. The relationship between forward (and backward) differences and the Khayyam-Pascal’, s triangle are also examined.

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).