Is it true that the sum of the k largest Laplacian eigenvalues of a graph with m edges is at most k(k+1)/2+m

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be a finite simple undirected graph on $n\geq1$ vertices with $m=|E(G)|$. Its adjacency matrix $A$ has $A_{ij}=1$ exactly when $i,j$ are adjacent, and $D=\operatorname{diag}(\deg_G(1),\ldots,\deg_G(n))$. The combinatorial Laplacian is $L=D-A$, whose real eigenvalues, with multiplicity, are ordered $\lambda_1\geq\cdots\geq\lambda_n$.

Statement recorded in the supplied source.

For every $G$ and every integer $1\leq k\leq n$, is \[\sum_{i=1}^k\lambda_i\leq m+\frac{k(k+1)}2?\] [S2]

Short English statement

Is the sum of the largest k Laplacian eigenvalues always at most the number of edges plus k(k+1)/2?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100031 (AMR-030-0031). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Joshua N. Cooper, Constraints on Brouwer's Laplacian Spectrum Conjecture (2020), section 1, equation (1), pp. 1--2. https://arxiv.org/abs/2003.03447.