A proof of the maximum Laplacian energy conjecture for connected graphs via a sharp eigenvalue-sum bound
Abstract
Let Sk(G) denote the sum of the k largest Laplacian eigenvalues of a connected graph G of order n and size m. Write PAn,ω for the graph obtained from an ω-vertex clique by attaching n−ω pendant vertices to one of its vertices, and set Mn,k:=binomk+12+n−k−1, the number of edges of PAn,k+1. For n/2<k≤ n−2, we prove the sharp bound Sk(G)≤ 2k/nm+ (2(n−k))/nMn,k−(n−k−1), with equality attained by PAn,k+1. This bound is complementary to Brouwer's inequality and is strictly stronger when m<Mn,k. Combining our bound with Brouwer's inequality, we resolve and strengthen a conjecture of Vinagre, Del-Vecchio, Justo, and Trevisan: for every n, the pineapple PAn,1+⌊2n/3⌋ maximizes the Laplacian energy among all connected graphs of order n; moreover, for n>4, it is the unique maximizer.
Related
- Connected graphs having maximum signless Laplacian spectral radius
- Eigenvalue comparison theorems for the Witten-Laplacian and the weighted p-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
- Sharp lower bound of the first eigenvalue of the p-Laplacian under integral Ricci curvature
- An endpoint exception to a conjectured equality characterization for Laplacian eigenvalue products
- Eigenvalue Distribution of the Laplacian on Random Complexes
BibTeX
@misc{mojallal2026proof,
title = {{A proof of the maximum Laplacian energy conjecture for connected graphs via a sharp eigenvalue-sum bound}},
author = {Seyed Ahmad Mojallal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06442},
eprint = {2609.06442},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06442},
}
RIS
TY - UNPB
TI - A proof of the maximum Laplacian energy conjecture for connected graphs via a sharp eigenvalue-sum bound
AU - Seyed Ahmad Mojallal
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06442
UR - https://arxiv.org/abs/2609.06442
AB - Let S_k(G) denote the sum of the k largest Laplacian eigenvalues of a connected graph G of order n and size m. Write PA_n,ω for the graph obtained from an ω-vertex clique by attaching n−ω pendant vertices to one of its vertices, and set M_n,k:=binomk+12+n−k−1, the number of edges of PA_n,k+1. For n/2<k≤ n−2, we prove the sharp bound S_k(G)≤ 2k/nm+ (2(n−k))/nM_n,k−(n−k−1), with equality attained by PA_n,k+1. This bound is complementary to Brouwer's inequality and is strictly stronger when m<M_n,k. Combining our bound with Brouwer's inequality, we resolve and strengthen a conjecture of Vinagre, Del-Vecchio, Justo, and Trevisan: for every n, the pineapple PA_n,1+⌊2n/3⌋ maximizes the Laplacian energy among all connected graphs of order n; moreover, for n>4, it is the unique maximizer.
ER -