Construction of graphs with distinct Aα-eigenvalues

Abstract

Let G be a connected graph of order n with adjacency matrix A(G) and degree diagonal matrix D(G). The Aα-matrix of G is defined as Aα(G)=αD(G)+(1−α)A(G) for any α∈[0,1). All the eigenvalues of Aα(G) are called Aα-eigenvalues of G, and the Aα-spectrum of G consists of all the Aα-eigenvalues along with their multiplicities. Denote the set of connected graphs (resp., of order n) with distinct Aα-eigenvalues by GAα (resp., GnAα). We also use GAα⁎ (resp., GnAα⁎) to denote the set of connected graphs in GAα (resp., GnAα) whose Aα-eigenvalues are main. Two graphs G and H are called Aα-cospectral if they have the same Aα-spectrum. This paper proposes a new approach to construct infinite families of GAα and GAα⁎. More specifically, for a graph G in GnAα or GnAα⁎, the infinite families of GAα or GAα⁎ are constructed from G. At the same time, the Aα-spectra of these graphs are completely determined by the Aα-spectrum of G. Finally, using this technique, we also construct some infinite families of non-isomorphic Aα-cospectral graphs in GAα and GAα⁎. Applying the results obtained to two cases of adjacency matrix and signless Laplacian matrix, we can derive the main results in [9] and [10].