Spectral supersaturation: Triangles and bowties

Recently, Ning and Zhai (2023) proved that every n-vertex graph G with λ (G) ≥ √ n²/4 has at least n/2 -1 triangles, unless G=Kn/2 , n/2. The aim of this paper is two-fold. Using the supersaturation-stability method, we prove a stability variant of Ning-Zhai's result by showing that such a graph G contains at least $n-3$ triangles if no vertex is in all triangles of G. This result could also be viewed as a spectral version of a result of Xiao and Katona (2021). The second part concerns with the spectral supersaturation for the bowtie, which consists of two triangles sharing a common vertex. A theorem of Erd{o}s, F\"{u}redi, Gould and Gunderson (1995) says that every n-vertex graph with more than n²/4 +1 edges contains a bowtie. For graphs of given order, the spectral supersaturation problem has not been considered for substructures that are not color-critical. In this paper, we give the first such theorem by counting the number of bowties. Let Kn/2 , n/2⁺² be the graph obtained from Kn/2 , n/2 by embedding two disjoint edges into the vertex part of size n/2. Our result shows that every graph G with n≥ 8.8 × 10⁶ vertices and λ (G)≥ λ (Kn/2 , n/2⁺²) contains at least n/2 bowties, and Kn/2 , n/2⁺² is the unique spectral extremal graph. This gives a spectral correspondence of a theorem of Kang, Makai and Pikhurko (2020). The method used in our paper provides a probable way to establish the spectral counting results for other graphs, even for non-color-critical graphs.