Sums of Laplacian eigenvalues and sums of degrees

Let $X$ be a simplicial complex. For $1\le i\le \dim (X)$, let $X(i)$ be the set of $i$-dimensional faces of $X$, and let $f_i(X)=\mathrm{\mid }X(i)\mathrm{\mid }$. For $0\le i\le \dim (X)-1$, let $L_i^{+}(X)$ be the $i$-th upper Laplacian operator of $X$. For $\sigma \in X$ and $1\le r\le \dim (X)$, we denote by ${\text{deg}}_X^{(r)}(\sigma )$ the number of $r$-dimensional faces of $X$ containing $\sigma$. For a symmetric matrix $M\in {\mathbb{R}}^{n\times n}$ and $1\le i\le n$, let ${\lambda }_i(M)$ be the $i$-th largest eigenvalue of $M$. We prove that for every complex $X$, $1\le r\le \dim (X)$, and $1\le k\le f_{r-1}(X)\mathrm{/}(r+1)$, $$\sum _{i=1}^k{\lambda }_i(L_{r-1}^{+}(X))\le \max \{\sum _{\sigma \in A}{\text{deg}}_X^{(r)}(\sigma ):A\subset X(r-1),\mathrm{\mid }A\mathrm{\mid }=(r+1)k\}\mathrm{.}$$ This bound is sharp, and it extends a classical result of Anderson and Morley, corresponding to the special case $k=1,r=1$. As a consequence, we show that for all $1\le r\le \dim (X)$ and $1\le k\le f_{r-1}(X)$, $$\sum _{i=1}^k{\lambda }_i(L_{r-1}^{+}(X))\le f_r(X)+\left(\genfrac{}{}{0px}{}{(r+1)k}{2}\right)\mathrm{.}$$ In the case $r=1$, we obtain the following improved bound: for every $k\ge 1$ and every graph $G=(V,E)$ with $\mathrm{\mid }V\mathrm{\mid }\ge k$, $$\sum _{i=1}^k{\lambda }_i(L(G))\le \mathrm{\mid }E\mathrm{\mid }+k^2,$$ where $L(G)=L_0^{+}(G)$ is the Laplacian matrix of $G$. This improves upon previously known bounds for all $k\ge 3$, and may be seen as a further step towards Brouwer’s conjecture, which states that ${\sum }_{i=1}^k{\lambda }_i(L(G))\le \mathrm{\mid }E\mathrm{\mid }+\left(\genfrac{}{}{0px}{}{k+1}{2}\right)\mathrm{.}$ As an additional application, we show that if $X$ is an $(r+1)$-partite $r$-dimensional simplicial complex on vertex set $V$, and $1\le k\le f_{r-1}(X)$, then $$\sum _{i=1}^k{\lambda }_i(L_{r-1}^{+}(X))\le \sum _{i=1}^k\mid \{v\in V:{\text{deg}}_X^{(r)}(v)\ge i\}\mid \mathrm{.}$$ This resolves a special case of a conjecture of Duval and Reiner, which states that the above inequality holds for all simplicial complexes.