On the Aα-spectral Radius of Graphs Without Large Matchings

Let G be an n -vertex graph. A matching in G is a set of independent edges, i.e., no two edges in the set are adjacent in G . The matching number is the maximal size of a matching in G . Nikiforov (Appl Anal Discrete Math 11(1):81–107, 2017) proposed the A α -matrix of a graph G , as follows: A α ( G ) = α D ( G ) + ( 1 - α ) A ( G ) , α ∈ [ 0 , 1 ] , where D ( G ) and A ( G ) are the diagonal degree matrix and adjacency matrix of G , respectively. In this contribution, we establish some A α -spectral conditions to guarantee that there do not exist large matchings in a graph G , recovering the previous results and obtaining similar results for a wide variety of spectral conditions from the A α -matrix. Our main tools are the Berge–Tutte formula and the double leading eigenvectors.