Spectra of Graphs with End Vertices Appended to All Vertices of the Base Graph: The Golden Ratio and Energy

In this paper, we determine the spectra of graphs obtained by appending h end vertex to all vertices of a defined class of graphs called the base graph. The end vertices allow for a quick solution to the eigen-vector equations satisfying the characteristic equation, and the solutions to the eigenvalues of the base graph arise. We determine the relationship between the eigenvalues of the base graph and the eigenvalues of the new graph as constructed above, and determine that if a is an eigenvalue of the base .........