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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 λ 1 , λ 2 , ... , λ 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 τ 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 Randić 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. [ABSTRACT FROM AUTHOR]

In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, we characterize the minimizing graph among all the connected graphs that belong to a class of graphs whose complements are bicyclic with two cycles. [ABSTRACT FROM AUTHOR]