@article{Alali2026, 
author = {Amal S. Alali and Kajal Rani and Shabir Ahmad Mir and Junaid Nisar},
title = {On a spectral concentration in Brouwer-type conjecture for a uniform caterpillar graphs},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {5},
pages = {15199-15214},
keywords = {distance Laplacian matrix, Wiener index, diameter, caterpillar graph},
url = {https://www.sciopen.com/article/10.3934/math.2026625},
doi = {10.3934/math.2026625},
abstract = {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  )  ≤  W  (  G  )  +                    (                              r          +          2                3                    )             holds for all    1  ≤  r  ≤  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      λ    1    (  G  ) form a strictly decreasing sequence for small values of    r and converge to a constant strictly less than one as    r  →  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.}
}