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arXiv · 2112.15279

Counting substructures and eigenvalues II: quadrilaterals

Bo Ning·Mingqing Zhai
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Abstract

Let $G$ be a graph and $\lambda(G)$ be the spectral radius of $G$. A previous result due to Nikiforov [Linear Algebra Appl., 2009] in spectral graph theory asserted that every graph $G$ on $m\geq 10$ edges contains a 4-cycle if $\lambda(G)>\sqrt{m}$. Define $f(m)$ to be the minimum number of copies of 4-cycles in such a graph. A consequence of a recent theorem due to Zhai et al. [European J. Combin., 2021] shows that $f(m)=\Omega(m)$. In this article, by somewhat different techniques, we prove that $f(m)=\Theta(m^2)$. We left the solution to $\lim\limits_{m\rightarrow \infty} \frac{f(m)}{m^2}$ as a problem, and also mention other ones for further study.

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Bo Ning, Mingqing Zhai. 2021-12-31. Counting substructures and eigenvalues II: quadrilaterals. https://doi.org/10.37236/12725

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