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

Amal S. Alali; Kajal Rani; Shabir Ahmad Mir; Junaid Nisar; Amal S. Alali; Kajal Rani; Shabir Ahmad Mir; Junaid Nisar

doi:10.3934/math.2026625

AIMS Mathematics

2026, Volume 11, Issue 5: 15199-15214. doi: 10.3934/math.2026625

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

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Lovely Professional University, Punjab 144411, India
3.
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
4.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India

Received: 17 March 2026 Revised: 11 May 2026 Accepted: 15 May 2026 Published: 29 May 2026
MSC : 05C12, 05C50

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) \leq W(G) + {r+2 \choose 3} $ holds for all $ 1\leq r \leq 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.
- distance Laplacian matrix,
- Wiener index,
- diameter,
- caterpillar graph
Citation: Amal S. Alali, Kajal Rani, Shabir Ahmad Mir, Junaid Nisar. On a spectral concentration in Brouwer-type conjecture for a uniform caterpillar graphs[J]. AIMS Mathematics, 2026, 11(5): 15199-15214. doi: 10.3934/math.2026625

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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) \leq W(G) + {r+2 \choose 3} $ holds for all $ 1\leq r \leq 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.

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