On Bounds of Energy of a Graph with Self-Loops

The energy of a graph GS with n vertices and self-loops is defined as, where the eigenvalues of the adjacency matrix of GS are 1, 2, . . ., n.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 1 < n, the quantity |i - n | is always greater than 0, 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.