On energies of graphs with given independence number and families of hyperenergetic graphs

Let G be a simple graph of order n and L (G) L^1 (G) its line graph. Then, the iterated line graph of G is defined recursively as L^2 (G) L (L (G) ), L^3 (G) L (L^2 (G) ), , L^k (G) (L^k-1 (G) ). The energy E (G) is the sum of absolute values of the eigenvalues of G. In this paper, it is derived a sharp upper bound for the energy of the line graph of a connected graph G of order n and independence number not less than where 1 n-2. This bound is attained, if and only if, G is isomorphic to the complete split graphs SK₍,. It is also determined a lower bound for the energy of the line graph of a graph G of order n and independence number. For 1 n-1 and H= (n-) K+1 (+-n) K, the equality holds, if and only if G H. As a consequence, families of hyperenergetic graphs are determined. Also, a lower bound for the energy of the iterated line of a graph G of order n and independence number is given and, for 1 n-1, the equality holds, if and only if, G K. Additionally, an upper bound for the incidence energy of connected graphs G of order n and independence number not less than is presented. Moreover, an upper bound on the Laplacian energy-like of the complement G of G is presented. For 1 n-1, the bound is attained, if and only if, G H. Finally, a Nordhaus-Gaddum type relation is given.