New Bounds for the α -Indices of Graphs

Let G be a graph, for any real 0 ≤ α ≤ 1 , Nikiforov defines the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , where A ( G ) and D ( G ) are the adjacency matrix and diagonal matrix of degrees of the vertices of G . This paper presents some extremal results about the spectral radius ρ α ( G ) of the matrix A α ( G ) . In particular, we give a lower bound on the spectral radius ρ α ( G ) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρ α ( G ) in terms of order and minimal degree. Furthermore, for n > l > 0 and 1 ≤ p ≤ ⌊ n − l 2 ⌋ , let G p ≅ K l ∨ ( K p ∪ K n − p − l ) be the graph obtained from the graphs K l and K p ∪ K n − p − l and edges connecting each vertex of K l with every vertex of K p ∪ K n − p − l . We prove that ρ α ( G p + 1 ) < ρ α ( G p ) for 1 ≤ p ≤ ⌊ n − l 2 ⌋ − 1 .