Abstract

Let G be a graph, for any real 0 <= alpha <= 1, Nikiforov defines the matrix A alpha(G) as A(alpha)(G) = alpha D(G)+(1 - alpha)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 rho(alpha)(G) of the matrix A(alpha)(G). In particular, we give a lower bound on the spectral radius rho(alpha)(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius rho(alpha)(G) in terms of order and minimal degree. Furthermore, for n > l > 0 and 1 <= p <= left perpendicularn-l/2right perpendicular, let G(p) congruent to to K-l boolean OR (K-p boolean OR Kn-p-l) be the graph obtained from the graphs K-l and K-p boolean OR Kn-p-l and edges connecting each vertex of K-l with every vertex of K-p boolean OR Kn-p-l. We prove that rho(alpha)(G(p+1)) < rho(alpha)(G(p)) for 1 <= p <= left perpendicularn-l/2right perpendicular - 1.