Structural and spectral analysis of Fibonacci graphs and their zagreb indices

This study explores the relationship between a specific type of graph and the Fibonacci sequence by introducing and analyzing the Fibonacci numbers graph, denoted as \(G_{f_n}\). We delve into the structural properties of \(G_{f_n}\) and establish new bounds for the first Zagreb index \(M_1(G_{f_n})\), relating it to the number of vertices n, the number of edges m, the maximum vertex degree \(\Delta\), the minimum vertex degree \(\delta\), and the clique number \(\omega\). Additionally, we investigate the domination number specific to Fibonacci graphs. Furthermore, we introduce two matrices: the equi-degree Laplacian matrix and the equi-degree signless Laplacian matrix, and examine their spectral characteristics to gain deeper insights into the eigenvalues of these matrices associated with connected graphs corresponding to Fibonacci numbers. This research not only broadens the theoretical understanding of Fibonacci graphs but also contributes to the field of algebraic graph theory by examining these new matrices.