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

  • Enide Andrade
  • Eber Lenes Universidad del Sin\'u
  • María Robbiano Universidad Católica del Norte
Keywords: Energy;, hyperenergetic graph, incidence energy, Laplacian energy-like, independence number, line graph, iterated line graph

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.

Author Biographies

Eber Lenes, Universidad del Sin\'u

Área de Ciencias Básicas Exactas

Grupo de Investigación Deartica

María Robbiano, Universidad Católica del Norte

Departamento de Matem\'aticas

Antofagasta

Published
2024-09-01
Section
Articles