PreprintCC BY-NC-ND 4.0arXiv
Supersaturation: Cliques and Positive Square Energy
Abstract
Let G be a simple graph of order n and let kr denote the number of unlabeled copies of Kr in G. Let s+ denote the sum of the squares of the positive adjacency eigenvalues of G. For every fixed integer r≥2, we prove that, as n → ∞ √(s+)>(1−1/r)n Longrightarrow kr+1≥ (1/(rr−1)−or(1))nr−1. The coefficient r−(r−1) is asymptotically best possible; in particular, the triangle case gives k3≥ n/2−o(n). The proof combines positive square energy stability, the clique removal lemma, and a sharp clique count around a maximum r-cut. The triangle case requires an additional block-matrix argument.
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BibTeX
@misc{jayarajan2026supersaturation,
title = {{Supersaturation: Cliques and Positive Square Energy}},
author = {Abhay Jayarajan and M. Rajesh Kannan and Shivaramakrishna Pragada},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40570},
eprint = {2610.40570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40570},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Supersaturation: Cliques and Positive Square Energy
AU - Abhay Jayarajan
AU - M. Rajesh Kannan
AU - Shivaramakrishna Pragada
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40570
UR - https://arxiv.org/abs/2610.40570
AB - Let G be a simple graph of order n and let k_r denote the number of unlabeled copies of K_r in G. Let s^+ denote the sum of the squares of the positive adjacency eigenvalues of G. For every fixed integer r≥2, we prove that, as n → ∞ √(s^+)>(1−1/r)n Longrightarrow k_r+1≥ (1/(r^r−1)−o_r(1))n^r−1. The coefficient r^−(r−1) is asymptotically best possible; in particular, the triangle case gives k_3≥ n/2−o(n). The proof combines positive square energy stability, the clique removal lemma, and a sharp clique count around a maximum r-cut. The triangle case requires an additional block-matrix argument.
ER -