On the spectral radius and energy of the weighted adjacency matrix of a graph

Let G be a graph of order n and let di be the degree of the vertex vi in G for i=1,2,…,n. The weighted adjacency matrix Adb of G is defined so that its (i, j)-entry is equal to di+djdidj if the vertices vi and vj are adjacent, and 0 otherwise. The spectral radius ϱ1 and the energy Edb of the Adb-matrix are examined. Lower and upper bounds on ϱ1 and Edb are obtained, and the respective extremal graphs are characterized.