Spectral supersaturation: Triangles and bowties

• Published in: European journal of combinatorics Vol. 128; p. 104171
• Main Authors: Li, Yongtao, Feng, Lihua, Peng, Yuejian
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.08.2025
• ISSN: 0195-6698
• DOI: 10.1016/j.ejc.2025.104171

A classical result of Erdős and Rademacher (1955) demonstrates a fundamental supersaturation phenomenon in extremal combinatorics: every graph on n vertices with more than ⌊n2/4⌋ edges contains at least ⌊n/2⌋ triangles. Let λ(G) be the spectral radius of the adjacency matrix of a graph G. Recently, Ning and Zhai (2023) proved that every n-vertex graph G with λ(G)≥⌊n2/4⌋ contains at least ⌊n/2⌋−1 triangles, unless G is a balanced complete bipartite graph K⌈n2⌉,⌊n2⌋. The aim of this paper is two-fold. Using a different approach which we term the supersaturation-stability method, we prove a stability variant of the Ning–Zhai result by showing that such a graph G contains at least n−3 triangles if no vertex lies in all triangles of G. This bound is the best possible and it could also be viewed as a spectral analogue of a theorem of Xiao and Katona (2021), which guarantees n−2 triangles under the assumption that e(G)>⌊n2/4⌋ and no vertex is in all triangles of G. The second part concerns with the spectral supersaturation for the bowtie, which consists of two triangles sharing a vertex. Erdős, Füredi, Gould and Gunderson (1995) proved that every n-vertex graph with more than ⌊n2/4⌋+1 edges contains a bowtie. The spectral supersaturation problem has not been investigated for non-color-critical substructures in graphs with given order. We give the first such result by counting bowties. Let K⌈n2⌉,⌊n2⌋+2 be the graph obtained from K⌈n2⌉,⌊n2⌋ by embedding two disjoint edges in the vertex part of size ⌈n2⌉. We show that if n≥8.8×106 and λ(G)≥λ(K⌈n2⌉,⌊n2⌋+2), then G has at least ⌊n2⌋ bowties, and K⌈n2⌉,⌊n2⌋+2 is the unique spectral extremal graph. This gives a spectral correspondence of a result of Kang, Makai and Pikhurko (2020). The method developed in our paper could be helpful in establishing the spectral results for counting other substructures, even for non-color-critical graphs.