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) ≡ L¹(G) its line graph. Then, the iterated line graph of G is defined recursively as L²(G) ≡ L(L(G)), L³(G)≡ L(L²(G)), …, Lᵏ(G)(Lᵏ⁻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 α where 1≤α≤ n-2. This bound is attained, if and only if, G is isomorphic to the complete split graphs SKn,α. It is also determined a lower bound for the energy of the line graph of a graph G of order n and independence number α. For 1≤α≤ n-1 and $H=(n-α/α)K/α+1 (α+α/α-n)K/α$, 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 $α$ is given and, for $1≤α≤ n-1$, the equality holds, if and only if, $G α K/α$. Additionally, an upper bound for the incidence energy of connected graphs $G$ of order $n$ and independence number not less than $α$ is presented. Moreover, an upper bound on the Laplacian energy-like of the complement $Ḡ$ of $G$ is presented. For $1≤α≤ n-1$, the bound is attained, if and only if, $G H.$ Finally, a Nordhaus-Gaddum type relation is given.