Bounds for the energy of weighted graphs

Let G be a simple connected graph with n vertices and m edges. Let W(G) = (G, w) be the weighted graph corresponding to G. Let lambda(1), lambda(2), ..., lambda(n) be the eigenvalues of the adjacency matrix A(W(G)) of the weighted graph W(G). The energy E(W(G)) of a weighted graph W(G) is defined as the sum of absolute value of the eigenvalues of W(G). In this paper, we obtain upper bounds for the energy E(W(G)), in terms of the sum of the squares of weights of the edges, the maximum weight, the maximum degree d(1), the second maximum degree d(2) and the vertex covering number tau of the underlying graph G. As applications to these upper bounds we obtain some upper bounds for the energy (adjacency energy), the extended graph energy, the Randie energy and the signed energy of the connected graph G. We also obtain some new families of weighted graphs where the energy increases with increase in weights of the edges. (C) 2019 Elsevier B.V. All rights reserved.