Improved matching-number bound for Laplacian eigenvalue sums

Let $G=(V,E)$ be a graph with $n$ non-isolated vertices, and let $\nu(G)$ denote the maximum size of a matching in $G$. Let $L(G)$ be the Laplacian matrix of $G$, with eigenvalues $\lambda_1(L(G))\geq\cdots\geq\lambda_n(L(G))$. Improved matching-number conjecture. If $1\leq k\leq n-2$, then

$$\sum_{i=1}^k \lambda_i(L(G))\leq |E|+k\cdot\nu(G).$$

The paper presents this as a stronger bound suggested by equality cases for a proved matching-number estimate. The restriction to $k\leq n-2$ is necessary for the complete graph of odd order, although the paper notes that it is not a significant restriction because the eigenvalue sum equals $2|E|$ for $k\geq n-1$.