Suppose $G$ is a simple graph of order $n$ with eigenvalues $\lambda_1\geq \cdots\geq \lambda_n$. I've encountered the quantity $L=\big\vert |\lambda_1|-|\lambda_2|-\cdots-|\lambda_n|\big\vert$. Note that if $\rho$ denotes the spectral radius of $G$ and $\mathcal{E}$ denotes the graph energy of $G$, then $$L=|2\rho-\mathcal{E}|.$$ I am no expert in spectral graph theory (or graph theory for that matter) but I was wondering if anybody had encountered this quantity before. Ultimately, I'm interested in lower bounds for $L$. Can one bound $L$ from below for all graphs? Perhaps if one restricts to a class of graphs? Thanks in advance!
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