On a distance Laplacian analog of Brouwer's conjecture for several classes of graphs

Abstract

Zhou et al. (2025) proposed a distance Laplacian analog of Brouwer's conjecture on partial sums of Laplacian eigenvalues, asserting that for any connected graph $G$, ${\sum }_{i=1}^r{\partial }_i^L(G)\le W(G)+\left(\genfrac{}{}{0ex}{}{r+2}{3}\right),$ where ${\partial }_i^L(G)$ are the eigenvalues of the distance Laplacian matrix and $W(G)$ is the Wiener index. We prove this inequality for three broad classes of graphs, thereby improving and extending existing results. First, we prove that all connected graphs of diameter at most $D$ satisfy the inequality once the order $n$ satisfies $n\ge \lceil \frac{4}{9}(D+1{)}^3\rceil$. Second, we show that the inequality holds for every diameter-$2$ graph with the only exceptions being $K_{1,3}$ at $r=2$ and $K_{1,4}$ at $r=3$. Third, we prove that if the maximum degree is $\Delta (G)=n-k$, then the inequality holds for all $n\ge N(k)$, where $N(2)=10$ and $N(k)=\lceil 5(k-1{)}^{3/2}\rceil$ for $k\ge 3$. Our proofs rely on decomposing the distance Laplacian matrix into Laplacian matrices of auxiliary graphs whose edges are vertex pairs at distance at least a prescribed value, together with classical eigenvalue inequalities.