The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. In this article we consider the problem whether generalized Fibonacci constants $varphi_n$ $(ngeq 2)$ can be the energy of graphs. We show that $varphi_n$ cannot be the energy of graphs. Also we prove that all natural powers of $varphi_{2n}$ cannot be the energy of a matroid.

The aim of this paper is to give new results about factorizations of the Fibonacci numbers Fn and the Lucas numbers Ln. These numbers are defined by the second order recurrence relation an+2 = an+1+an with the initial terms F0 = 0, F1 = 1 and L0 = 2, L1 = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and ...

We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions avoiding certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2, Πn(13/2), and those avoiding both 13/2 and 123, Πn(13/2, 123). We show that the distribution over Πn(13/2) enumerates certain integer partitions, and the distr...

In a recent paper, Goyt and Sagan studied distributions of certain set partition statistics over pattern restricted sets of set partitions that were counted by the Fibonacci numbers. Their study produced a class of q-Fibonacci numbers, which they related to q-Fibonacci numbers studied by Carlitz and Cigler. In this paper we will study the distributions of some Mahonian statistics over pattern r...

We define the convolved hðxÞ-Fibonacci polynomials as an extension of the classical con-volved Fibonacci numbers. Then we give some combinatorial formulas involving the hðxÞ-Fibonacci and hðxÞ-Lucas polynomials. Moreover we obtain the convolved hðxÞ-Fibo-nacci polynomials from a family of Hessenberg matrices. Fibonacci numbers and their generalizations have many interesting properties and appli...

In this paper, we introduce the h-analogue of Fibonacci numbers for non-commutative h-plane. For hh = 1 and h = 0, these are just the usual Fibonacci numbers as it should be. We also derive a collection of identities for these numbers. Furthermore, h-Binet’s formula for the h-Fibonacci numbers is found and the generating function that generates these numbers is obtained. 2000 Mathematical Subje...

In this article, the Apostol Bernoulli-Fibonacci polynomials are defined and various properties of obtained. Furthermore, Euler-Fibonacci numbers found. addition, harmonic-based F exponential generating functions for numbers.

We give the bicomplex Gaussian Fibonacci and Lucas numbers establish generating functions Binet’s formulas related to these numbers. Also, we present summation formula, matrix representation Honsberger identity their relationship between Finally, show relationships among Fibonacci, Lucas,