Combinatorics Of Fibonacci And Lucas Cube Polynomials

The study of the graph polynomials is an important topic in graph theory, it set up a bridge between graph theory and algebra. Not only the polynomial of graphs contains numerous information about graphs, but also the coefficient sequence contains a rich combination of knowledge. The research on the topic includes the study of unimodality, log-concavity, log-convexity, the reality of zeros and a series of problems, the thesis is focused on some research of the Fibonacci and Lucas cube polynomials, such as, the explicit expression of the two kinds of cube polynomials, q-log-concavity (convexity) of these two polynomial sequences, the relevant equations about them, and some related results are released. The thesis is organized as follows.In the first chapter, we introduce some basic concepts and conclusions used in the thesis and gives a brief description of graph polynomials.In the second chapter, we study the Fibonacci and Lucas cube polynomials. Based on Klavvzar and Mollard’s studies, trigonometric function expressions of the two polynomials are obtained according to algebraic methods. The reality of zeros of them are described from a new point of view, naturally, the unimodality and log-concavity are acquired.In the third chapter, we investigate the q-log-convexity and q-log-concavity of Fi-bonacci and Lucas cube polynomial sequences. According to the second chapter and the recurrence formulas of them, q-log-convexity and q-log-concavity are related to the parity of n. And on this basis, some relevant deformation formulas and identities are obtained about the Fibonacci and Lucas cube polynomials.