A characterization of equality in the edge-localized Bollobás-Nikiforov inequality
Abstract
Let G be a simple graph of order n≥2. Let e(G) and ω(G) denote its number of edges and clique number, respectively. Let λ1(G)≥⋯≥λn(G) be its adjacency eigenvalues. In 2007, Bollobás and Nikiforov conjectured that every graph G≠ Kn satisfies λ12(G)+λ22(G)≤2(1−1/ω(G))e(G). This conjecture was recently completely resolved by Coutinho, Liu, Spier, Tang and Zhang. For every edge e∈ E(G), let cG(e) denote the maximum order of a clique containing e. Kannan, Kumar and Pragada proposed the edge-localized conjecture λ12(G)+λ22(G)≤2∑e∈ E(G)(1−1/cG(e)) for every graph G≠ Kn. This conjecture was subsequently proved by Shaik. In this paper, we provide complete structural characterizations of the equality cases in both the Bollobás-Nikiforov inequality and its edge-localized version.
Related
- A Robust Bollobás-Nikiforov Bound for Graphs Close to Triangle-Free Graphs
- A new proof of equality cases of the Brunn-Minkowski inequality on manifolds
- Jensen' s trace inequality with equality condition and noncommutative Lamperti's theorem
- On the equality case in the Kotrbatý-Mouamine inequality
- Equality in the Bosonic Quantum Entropy Power Inequality
BibTeX
@misc{zhang2026characterization,
title = {{A characterization of equality in the edge-localized Bollobás-Nikiforov inequality}},
author = {Chuling Zhang and Xiying Yuan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36912},
eprint = {2610.36912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36912},
}
RIS
TY - UNPB
TI - A characterization of equality in the edge-localized Bollobás-Nikiforov inequality
AU - Chuling Zhang
AU - Xiying Yuan
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36912
UR - https://arxiv.org/abs/2610.36912
AB - Let G be a simple graph of order n≥2. Let e(G) and ω(G) denote its number of edges and clique number, respectively. Let λ_1(G)≥⋯≥λ_n(G) be its adjacency eigenvalues. In 2007, Bollobás and Nikiforov conjectured that every graph G≠ K_n satisfies λ_1^2(G)+λ_2^2(G)≤2(1−1/ω(G))e(G). This conjecture was recently completely resolved by Coutinho, Liu, Spier, Tang and Zhang. For every edge e∈ E(G), let c_G(e) denote the maximum order of a clique containing e. Kannan, Kumar and Pragada proposed the edge-localized conjecture λ_1^2(G)+λ_2^2(G)≤2∑_e∈ E(G)(1−1/c_G(e)) for every graph G≠ K_n. This conjecture was subsequently proved by Shaik. In this paper, we provide complete structural characterizations of the equality cases in both the Bollobás-Nikiforov inequality and its edge-localized version.
ER -