The Sharp Upper Bounds on the Aα-Spectral Radius of C4-Free Graphs and Halin Graphs.

Let G be a simple undirected graph. For any real number α ∈ [ 0 , 1 ] , Nikiforov defined the A α -matrix of G as A α (G) = α D (G) + (1 - α) A (G) , where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. The largest eigenvalue of A α (G) is called the A α -spectral radius of G. In this paper, we give sharp upper bounds on the A α -spectral radius of C 4 -free graphs and Halin graphs for α ∈ [ 1 / 2 , 1) respectively. [ABSTRACT FROM AUTHOR]