05C50 Online Seminar: Carla Oliveira

Let G = (V, E) be a simple graph of order n. The adjacency matrix of G is denoted by A(G) = [a_ij], where a_ij=1 if {v_i ,v_j} ∈ E and a_ij=0, otherwise. The diagonal matrix of degrees of G, D(G) = [d_ij], is defined by d_ii=d(v_i), and d_ij=0, ∀ i \neq j, and the signless Laplacian matrix of G is defined as Q(G) = D(G) + A(G). In 2017, Nikiforov define the matrix Aα(G) as a convex linear combination of A(G) and D(G) as follows Aα(G)= αD(G) + (1 − α)A(G), α ∈ [0, 1]

It is easy to see that A_0(G) = A(G), A_1(G) = D(G) and 2A_{1/2}(G) = Q(G). This talk presents some results and problems involving the eigenvalues of the Aα-matrix.