On a spectral concentration in Brouwer-type conjecture for a uniform caterpillar graphs

In this paper, we investigate spectral properties of the distance Laplacian matrix of certain graphs. We derive bounds on the distance Laplacian eigenvalues of a uniform caterpillar graphs and establish Brouwer-type inequalities for a graph with sufficiently large diameter. We verify the Brouwer-type conjecture proposed by Zhou et al. for the class of uniform caterpillar graphs with diameter at least six, thereby confirming its validity for a new infinite family of trees. The inequality $U_r(G)\le W(G)+(\genfrac{}{}{0ex}{}{r+2}{3})$ holds for all $1\le r\le n-1,$ where $U_r(G)$ is the sum of the $r$ largest distance Laplacian eigenvalues, and $W(G)$ is the Wiener index. Moreover, we show that the normalized spectral sums $U_r(G)/r{\lambda }_1(G)$ form a strictly decreasing sequence for small values of $r$ and converge to a constant strictly less than one as $r\to n-1$, revealing a spectral compression phenomenon in the distance Laplacian spectrum. Several analytical bounds are provided to demonstrate the tightness of the obtained results.