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

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)+\binom{r+2}{3} $ holds, 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\frac49(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 some classical eigenvalue inequalities.