2017/04/22 by Óscar Rojo, Rojo, Oscar
Mathematics · Chemistry · #Graph theory and applications #Finite Group Theory Research #Molecular spectroscopy and chirality
paper · pdf · doi:10.48550/arxiv.1704.06730
Given a connected graph R on r vertices and a rooted graph H, let R\H\ be the graph obtained from r copies of H and the graph R by identifying the root of the i-th copy of H with the i-th vertex of R. Let 0≤α≤1, and let Aα(G)=αD(G)+(1-α)A(G) where D(G) and A(G) are the diagonal matrix of the vertex degrees of G and the adjacency matrix of G, respectively. A basic result on the Aα- spectrum of R\H\ is obtained. This result is used to prove that if H=Bk is a generalized Bethe tree on k levels, then the eigenvalues of Aα(R\Bk\) are the eigenvalues of symmetric tridiagonal matrices of order not exceeding k; additionally, the multiplicity of each eigenvalue is determined. Finally, applications to a unicyclic graph are given, including an upper bound on the α- spectral radius in terms of the largest vertex degree and the largest height of the trees obtained by removing the edges of the unique cycle in the graph.