Spectral Dynamics of Graph Sequences Generated by Subdivision and Triangle Extension

For a graph G and a unary graph operation X, there is a graph sequence {Gk} generated by G0 = G and Gk+1 = X(Gk). Let Sp(Gk) denote the set of normalized Laplacian eigenvalues of Gk. The set of limit points of ⋃∞ k=0 Sp(Gk), lim infk→∞ Sp(Gk) and lim supk→∞ Sp(Gk) are considered in this paper for graph sequences generated by two operations: subdivision and triangle extension. It is obtained that the spectral dynamic of graph sequence generated by subdivision is determined by a quadratic function, which is closely related to the the well-known logistic map; while that generated by triangle extension is determined by a linear function. By using the knowledge of dynamic system, the spectral dynamics of graph sequences generated by these two operations are characterized. For example, it is found that, for any initial non-trivial graph G, chaos takes place in the spectral dynamics of iterated subdivision graphs, and the set of limit points is the entire closed interval [0, 2].