Some Bounds on the Energy of Graphs with Self-Loops regarding ${\lambda }_1$ and ${\lambda }_n$

Abstract

Let $G_S$ be a graph with $n$ vertices obtained from a simple graph $G$ by attaching one self-loop at each vertex in $S\subseteq V(G)$. The energy of $G_S$ is defined by Gutman et al. as $E(G_S)={\sum }_{i=1}^n\left|{\lambda }_i-\frac{\sigma }{n}\right|$, where ${\lambda }_1,\dots ,{\lambda }_n$ are the adjacency eigenvalues of $G_S$ and $\sigma$ is the number of self-loops of $G_S$. In this paper, several upper and lower bounds of $E(G_S)$ regarding ${\lambda }_1$ and ${\lambda }_n$ are obtained. Especially, the upper bound $E(G_S)\le \sqrt{n\left(2m+\sigma -\frac{{\sigma }^2}{n}\right)}$ $(\ast )$ given by Gutman et al. is improved to the following bound \begin{align*} E(G_{S})\leq \sqrt{n\left(2m+σ-\frac{σ^{2}}{n}\right)-\frac{n}{2}\left(\left |λ_{1}-\fracσ{n}\right |-\left |λ_{n}-\fracσ{n}\right |\right)^{2}}, \end{align*} where $\left|{\lambda }_1-\frac{\sigma }{n}\right|\ge \cdots \ge \left|{\lambda }_n-\frac{\sigma }{n}\right|$. Moreover, all graphs are characterized when the equality holds in Gutmans' bound $(\ast )$ by using this new bound.