Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\binom{k+1}{2}$ for every $k\in \{1,2,\dots,n\}$, which has been confirmed by Kothari and Tudose (2026) recently. In this note, we characterize the equality case in this inequality. Our main result is that for every $n$-vertex graph $G=(V,E)$ and for every $k\in \{1,2,\dots,n-1\}$, the equality $\sum_{i=1}^k\mu_i(G)=|E(G)|+\binom{k+1}{2}$ holds if and only if $G$ is a threshold graph with clique number $k+1$, where $\mu_1(G)\geq \mu_2(G)\geq \cdots\geq \mu_{n}(G)$ are the Laplacian eigenvalues of $G$. This, together with the confirmed Brouwer's conjecture, would yield a complete solution to the full Brouwer's conjecture posed by Li and Guo (2022). Our proof relies on the projection method of Kothari and Tudose and shows directly that the equality case can occur only for threshold graphs.
For a simple graph $G$ of order $n$, let $S_2(G)=\lambda_1(G)+\lambda_2(G)$ denote its spectral sum. We determine, for every $n\geq5$, the exact maximum of $S_2(G)$ and all equality cases. The unique maximizer, up to isomorphism, is the complement of the disjoint union of a suitably balanced complete bipartite graph and isolated vertices, with the sizes of its three parts determined by $n$ modulo $7$. Denoting this graph by $K_n^\star$, we further show that $ S_2(K_n^\star)\leq\frac{8n}{7}-2,$ with equality exactly when $7\mid n$. This proves a conjecture of Kumar, Liu, Monterde, Pragada and Tait, which strengthens the Aouchiche--Hansen 2010 conjecture by extending it from connected graphs to all graphs and by asserting uniqueness of the extremal graph. The result also subsumes the 2008 conjecture of Ebrahimi B., Mohar, Nikiforov, and Ahmady. The proof combines Ky Fan's variational principle with a spectral inequality for weighted Ferrers quotients to reduce the problem to an explicit family whose complements have incidence rank one. Exact integer optimization and a separate equality analysis then yield the maximum and uniqueness.
For an integer $k\ge2$, let $\lambda_k(G)$ denote the $k$th largest adjacency eigenvalue of a graph $G$. For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$, we prove \[ \lambda_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1. \] Our bound is tight for $k\in\{2,3,4,8,24\}$. We obtain it by reducing the graph-eigenvalue problem to an extremal problem for orthogonal projections and then applying the general upper bound on the absolute projection constant $\gamma(r)$ due to Der\k{e}gowska and Lewandowska. We also give an alternative proof of their bound by repairing the Gegenbauer-polynomial argument of K\"onig and Tomczak-Jaegermann. The resulting slack identity yields a strict improvement in every even dimension $r\ge4$ for which $r+2$ is not a perfect square.
For a simple graph $G$ of order $n$, let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ denote its adjacency eigenvalues. Hong's problem asks for the optimal upper bound for $\lambda_k(G)$. A recent theorem of Sivashankar gives, for every $k\ge3$, \[ \lambda_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1, \] with sharp examples arising from maximal real equiangular tight frames. In this paper, we characterize the equality case. We also obtain an explicit combinatorial description of the extremal graphs for $\lambda_3$ and $\lambda_4$.
Hitesh Kumar, Bojan Mohar, S. A. Mojallal et al.· 0 citations
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 sum of the largest Laplacian eigenvalues of graphs, {\em Discrete Appl. Math.}, 170:95--103, (2014)], Rocha and Trevisan showed that the conjecture holds true for $1 \leq k \leq \lfloor g/5 \rfloor$, where $g$ denotes the girth of $G$. This bound on $k$ was later improved by Chen in [Improved results on Brouwers conjecture for sum of the Laplacian eigenvalues of a graph, {\em Linear Algebra Appl.}, 557:327--338, (2018)], who established that the conjecture holds for $1 \leq k \leq \lfloor g/4 \rfloor$. In this article, we further strengthen these results by proving that the Brouwers conjecture holds for $1\leq k\leq\left\lfloor \frac{m(G)}{2}\right\rfloor,$ where $m(G)$ denotes the matching number of $G$. Since $m(G)\geq \lfloor g/2\rfloor$, the case constitutes a genuine improvement over the aforementioned results. As an application, we show that if $G$ is a graph of order $n$ and with a perfect matching, then the Brouwers conjecture holds for $1\leq k\leq \left\lfloor \frac{n}{4}\right\rfloor$. Finally, we provide a new perspective on verifying Brouwers conjecture by proving that $G$ satisfies the Brouwers conjecture if and only if for a fixed positive integer $h$, $\mathcal{S}_h(\overline{G})\leq e(\overline{G})+\binom{h+1}{2}$ holds whenever $\mathcal{S}_h(G)\leq e(G)+\binom{h+1}{2}$, where $\overline{G}$ denotes the complement of $G$.
The Grone--Merris inequality, conjectured by Grone and Merris~(1994) and first proved by Bai~(2011), states that for every graph $G$ of order $n$ and every $1\le k\le n$, $\sum_{i=1}^k\lambda_i(G)\le\sum_{i=1}^k d_i^*(G)$, where $\lambda_1\ge\cdots\ge\lambda_n$ are the Laplacian eigenvalues and $d_1^*\ge\cdots\ge d_n^*$ is the conjugate degree sequence. In this paper we determine exactly when equality holds. Using the split-graph trace inequality developed by Kothari and Tudose~(2026) in their proof of Brouwer's Laplacian conjecture---which relies on Bai's theorem and also establishes the equivalence between the two conjectures---together with the recent characterization of the Brouwer equality cases by Cai, Chen, Yang and Zhang~(2027), we prove that equality holds in the Grone--Merris inequality if and only if the graph $G$ belongs to one of two explicitly described families. Both families are obtained from a threshold graph by a surgical operation at one terminal block: in the first family, edges are removed from the initial dominating block; in the second, edges are added inside the initial isolated block. Our analysis yields a complete combinatorial description of all pairs $(G,k)$ for which the Grone--Merris bound is tight.
Dong-Xiu Cai, Zheng-Bo Chen, Jia Yang et al.· 1 citation
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.
S. A. Mojallal· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
