Abstract
The energy of a graph with vertices and self-loops is defined as , where the eigenvalues of the adjacency matrix of are . In this article, we have established some upper and lower bounds for the energy of such a graph. Those new bounds involve parameters like number of vertices (n), number of edges (m), number of self-loops (, maximum vertex degree (), and minimum vertex degree (). We show that for , the quantity is always greater than , and using that fact we establish a lower bound. We have compared and concluded that the new bounds are either better than the existing bounds or incomparable to a
few bounds obtained by some researchers recently.