Construction of graphs with distinct eigenvalues

Let G (resp. Gn) be the set of connected graphs (resp. with n vertices) whose eigenvalues are mutually distinct, and G∗ (resp. Gn∗) the set of connected graphs (resp. with n vertices) whose eigenvalues are mutually distinct and main. Two graphs G and H are said to be cospectral if they share the same adjacency spectrum. In this paper, we give a new method to construct infinite families of graphs in G and G∗. Concretely, given a graph G in Gn or Gn∗, the infinite families of G or G∗ are constructed from G, and furthermore the spectra of such graphs are also characterized by the spectrum of G. By the way, we use this method to construct some infinite families of non-isomorphic cospectral graphs, especially, including the graphs in G and G∗.