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 an...
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 grap...
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 exam...
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 su...
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^*...
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...
S. A. Mojallal· 0 citations
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