Lower Bounds for Energy of Matrices and Energy of Regular Graphs

Mohammad Reza Oboudi

Let $A=[a_{ij}]$ be an $n×n$ real symmetric matrix with eigenvalues $\lambda_1,\ldots,\lambda_n$. The energy of $A$, denoted by $\E(A)$, is defined as $|\lambda_1|+\cdots+|\lambda_n|$. We prove that if $A$ is non-zero and $|\lambda_1|\geq\cdots\geq|\lambda_n|$, then \begin{align}abel{corona} \E(A)\geq\frac{n|ambda_1||ambda_n|+um_{1eq i,jeq n}a^2_{ij}}{|ambda_1|+|ambda_n|}. \end{align} In particular, we show that $\Psi(A)\E(A)\geq\sum_{1\leq i,j\leq n}a^2_{ij},$ where $\Psi(A)$ is the maximum value of the sequence $\sum_{j=1}^{n}|a_{1j}|,\sum_{j=1}^{n}|a_{2j}|,\ldots,\sum_{j=1}^{n}|a_{nj}|$. The energy of a simple graph $G$, denoted by $\E(G)$, is defined as the energy of its adjacency matrix. As an application of inequality~(\ref{corona}) we show that if $G$ is a $t$-\,regular graph ($t\neq0$) of order $n$ with no eigenvalue in the interval $(-1,1)$, then $\E(G)\geq\frac{2tn}{t+1}$ and the equality holds if and only if every connected component of $G$ is the complete graph $K_{t+1}$ or the crown graph $K^{\star}_{t+1}$.