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

Abstract

 Let $G$ be a simple graph of order $n$ and $\mathscr{L}(G) \equiv \mathscr{L}^{1}(G)$ its line graph.

Then, the iterated line graph of $G$ is defined recursively as $\mathscr{L}^{2}(G) \equiv \mathscr{L}(\mathscr{L}(G)), \mathscr{L}^{3}(G)\equiv \mathscr{L}(\mathscr{L}^{2}(G)), \ldots, \mathscr{L}^{k}(G)\equiv\mathscr{L}\left(\mathscr{L}^{k-1}(G)\right).$ The energy $\mathcal{E}(G)$ 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 $\alpha$ where $1\leq\alpha\leq n-2$. This bound is attained, if and only if, $G$ is isomorphic to the complete split graphs $SK_{n,\alpha}$. It is also determined a lower bound for the energy of the line graph of a graph $G$ of order $n$ and independence number $\alpha$. For $1\leq\alpha\leq n-1$ and $\mathcal{H}=\left(n-\alpha\left\lfloor\dfrac{n}{\alpha}\right\rfloor\right)K_{\lfloor\frac{n}{\alpha}\rfloor+1}\bigcup \left(\alpha+\alpha\left\lfloor\dfrac{n}{\alpha}\right\rfloor-n\right)K_{\lfloor\frac{n}{\alpha}\rfloor}$, the equality holds, if and only if $G \cong \mathcal{H}.$

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 $\alpha$  is given and, for $1\leq\alpha\leq n-1$, the equality holds, if and only if, $G\cong \alpha K_{\left\lfloor\frac{n}{\alpha}\right\rfloor}$. Additionally, an upper bound for the incidence energy of connected graphs $G$ of order $n$ and independence number not less than $\alpha$ is presented. Moreover, an upper bound on the Laplacian energy-like of the complement $\overline{G}$ of $G$ is presented. For $1\leq\alpha\leq n-1$, the bound is attained, if and only if, $G\cong \mathcal{H}.$ Finally, a Nordhaus-Gaddum type relation is given.