A proof of the maximum Laplacian energy conjecture for connected graphs via a sharp eigenvalue-sum bound

Let $S_k(G)$ denote the sum of the $k$ largest Laplacian eigenvalues of a connected graph $G$ of order $n$ and size $m$. Write ${\mathrm{P}\mathrm{A}}_{n,\omega }$ for the graph obtained from an $\omega$-vertex clique by attaching $n-\omega$ pendant vertices to one of its vertices, and set $$M_{n,k}:=\left(\genfrac{}{}{0px}{}{k+1}{2}\right)+n-k-1,$$ the number of edges of ${\mathrm{P}\mathrm{A}}_{n,k+1}$. For $n\mathrm{/}2<k\le n-2$, we prove the sharp bound $$S_k(G)\le \frac{2k}{n}m+\frac{2(n-k)}{n}{M}_{n,k}-(n-k-1),$$ with equality attained by ${\mathrm{P}\mathrm{A}}_{n,k+1}$. This bound is complementary to Brouwer’s inequality and is strictly stronger when $m<M_{n,k}$. Combining our bound with Brouwer’s inequality, we resolve and strengthen a conjecture of Vinagre, Del-Vecchio, Justo, and Trevisan: for every $n$, the pineapple ${\mathrm{P}\mathrm{A}}_{n,1+\lfloor 2n\mathrm{/}3\rfloor }$ maximizes the Laplacian energy among all connected graphs of order $n$; moreover, for $n>4$, it is the unique maximizer.