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 . [ABSTRACT FROM AUTHOR]