In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field ${\mathbb{Z}}_p$ for special values of prime integer p.

In this paper, we present a further generalization of the bi- periodic Fibonacci quaternions and octonions. We give the generating function, the Binet formula, and some basic properties of these quaternions and octonions. The results of this paper not only give a generalization of the bi-periodic Fibonacci quaternions and octonions, but also include new results such as the matrix representation and the norm value of the generalized bi-periodic Fibonacci quaternions.

In this paper, a new graph type similar to binomial graphs is constructed using fibonomial coefficients. The spectrum of this new graph was obtained, the energy of the graph and the sum of the Laplacian eigenvalues are calculated. In addition, the connectivity feature of the graph is examined and the properties of the vertices forming the graph are revealed.

Many generalizations have been made for Fibonacci and Lucas number sequences and many properties have been found about these sequences. In the article [13], the authors obtained many features of these sequences with the Cholesky decomposition algorithm, using the 2 x 2 matrix belonging to a generalization of the Fibonacci sequence. In this study, it is shown that many different features can be found by using a 2 x 2 matrix belonging to the Lucas number sequence with the same method.