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 $\mathrm{PA}_{n,\omega}$ for the graph obtained from an $\omega$-vertex clique by attaching $n-\omega$ pendant vertices to one of its vertices, and set \[ M_{n,k}:=\binom{k+1}{2}+n-k-1, \] the number of edges of $\mathrm{PA}_{n,k+1}$. For $n/2<k\le n-2$, 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 $\mathrm{PA}_{n,k+1}$. 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 $\mathrm{PA}_{n,1+\lfloor2n/3\rfloor}$ maximizes the Laplacian energy among all connected graphs of order $n$; moreover, for $n>4$, it is the unique maximizer.
Let $B_{n,k}$ be the graph obtained from the balanced complete bipartite graph on $n$ vertices by deleting a matching of size $k$. If $G$ is an $n$-vertex triangle-free graph with $\kappa(\comp G)\geq k$, we prove that $\rhoA(G)\leq\rhoA(B_{n,k})$ for $n\geq4k+2$, with equality precisely when $G\cong B_{n,k}$, and we c...
For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1}...
Let $G$ be a simple graph on $n$ vertices and $e(G)$ edges. Let $\mu_1\geq \cdots \geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of $G$. For $k=1, \ldots, n$, let $S_k(G)=\sum_{i=1}^{k}\mu_i$. Brouwers conjecture asserts that for any $k\in\{1,\ldots, n\}$, $S_k(G)\leq e(G)+\binom{k+1}{2}$. In [Bounding the su...
Let $\Sigma=(G,\sigma)$ be a connected signed graph of order $n$ and size $m$, and let $s^{+}(\Sigma)$ and $s^{-}(\Sigma)$ denote the sums of the squares of its positive and negative adjacency eigenvalues, respectively. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan states that every connected gr...
For a connected graph $G$, let $\tau(G)$ denote the maximum number of pairwise edge-disjoint spanning trees, and let $\rho(G)$ be its adjacency spectral radius. For integers $n\ge1$, $k\ge2$, and $k\le\delta\le2k-1$, let $\mathcal{G}_{n,\delta}$ be the class of connected $n$-vertex graphs with minimum degree $\delta$,...
For a family $\mathcal{F}$ of $k$-graphs, $\ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $\ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n...
