LAUSR

Abstract Let G be a graph with adjacency matrix A ( G ) and with D ( G ) the diagonal matrix of its vertex degrees. Nikiforov defined the matrix A α ( G ) , with α ∈ [ 0 , 1 ] , as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . The largest and the smallest eigenvalues of A α ( G ) are respectively denoted by ρ ( G ) and λ n ( G ) . In this paper, we present a tight upper bound for ρ ( G ) in terms of the vertex degrees of G for α ≠ 1 . For any graph G, ρ ( G ) ≤ max u ∼ w ⁡ { α ( d u + d w ) + α 2 ( d u + d w ) 2 + 4 ( 1 − 2 α ) d u d w 2 } . If G is connected, for α ∈ [ 0 , 1 ) , the equality holds if and only if G is regular or bipartite semi-regular. As an application, we solve the problem raised by Nikiforov in [15] . For the smallest eigenvalue λ n ( G ) of a bipartite graph G of order n with no isolated vertices, for α ∈ [ 0 , 1 ) , then λ n ( G ) ≥ min u ∼ w ⁡ { α ( d u + d w ) − α 2 ( d u + d w ) 2 + 4 ( 1 − 2 α ) d u d w 2 } . If G is connected and α ≠ 1 / 2 , the equality holds if and only if G is regular or semi-regular.