A Lower (Upper) Bound for the Energy of Graphs

Let G be a graph of order n and size m. In this paper, we determine an upper bound for the energy of non-singular graph G in terms of order n, size m, positive and negative indices of inertia of A(G), and det(A(G)). We also obtain a lower bound for the energy of graph G, which relies on order n, size m, and maximum degree Δ. Furthermore, we identify extremal graphs that attain equality in each of these bounds.