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Proofs of Li, Wang and Guo's conjectures on (signless) Laplacian spectral radius
Abstract
Let μ(G) and q(G) be the Laplacian and signless Laplacian spectral radii of a graph G. Li, Wang and Guo (2026) conjectured that, if H with |V(H)|≥ 1 is an induced subgraph of a graph G, then μ(G)≤maxv∈ V(H)μ(Hvt) and q(G)≤maxv∈ V(H)q(Hvt), where t=e(G)−e(H) and Hvt is the graph obtained from H by adding t pendant edges to v∈ V(H). We prove this conjecture by establishing a spectral comparison theorem for a broader family of positive semidefinite graph matrices which includes the Laplacian and signless Laplacian matrices. We also characterize all equality cases for the prescribed subgraph H with e(H)≥ 1 and t≥ 1.
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BibTeX
@misc{chen2026proofs,
title = {{Proofs of Li, Wang and Guo's conjectures on (signless) Laplacian spectral radius}},
author = {Hongzhang Chen and Jianxi Li and Xiaoyun Lv},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29048},
eprint = {2610.29048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29048},
}
RIS
TY - UNPB
TI - Proofs of Li, Wang and Guo's conjectures on (signless) Laplacian spectral radius
AU - Hongzhang Chen
AU - Jianxi Li
AU - Xiaoyun Lv
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29048
UR - https://arxiv.org/abs/2610.29048
AB - Let μ(G) and q(G) be the Laplacian and signless Laplacian spectral radii of a graph G. Li, Wang and Guo (2026) conjectured that, if H with |V(H)|≥ 1 is an induced subgraph of a graph G, then μ(G)≤max_v∈ V(H)μ(H_v^t) and q(G)≤max_v∈ V(H)q(H_v^t), where t=e(G)−e(H) and H_v^t is the graph obtained from H by adding t pendant edges to v∈ V(H). We prove this conjecture by establishing a spectral comparison theorem for a broader family of positive semidefinite graph matrices which includes the Laplacian and signless Laplacian matrices. We also characterize all equality cases for the prescribed subgraph H with e(H)≥ 1 and t≥ 1.
ER -