Induced subgraphs of product graphs and a generalization of Huang's theorem

Recently, Huang showed that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional hypercube has maximum degree at least $\sqrt{n}$ in [Annals of Mathematics, 190 (2019), 949--955]. In this paper, we discuss the induced subgraphs of Cartesian product graphs and semi-strong product graphs to generalize Huang's result. Let $\Gamma_1$ be a connected signed bipartite graph of order $n$ and $\Gamma_2$ be a connected signed graph of order $m$. By defining two kinds of signed product of $\Gamma_1$ and $\Gamma_2$, denoted by $\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma_1\widetilde{\bowtie} \Gamma_2$, we show that if $\Gamma_1$ and $\Gamma_2$ have exactly two distinct adjacency eigenvalues $\pm\theta_1$ and $\pm\theta_2$ respectively, then every $(\frac{1}{2}mn+1)$-vertex induced subgraph of $\Gamma_1\widetilde{\Box}\Gamma_2$ (resp. $\Gamma_1\widetilde{\bowtie} \Gamma_2$) has maximum degree at least $\sqrt{\theta_1^2+\theta_2^2}$ (resp. $\sqrt{(\theta_1^2+1)\theta_2^2}$). Moreover, we discuss the eigenvalues of $\Gamma_1\widetilde{\Box} \Gamma_2$ and $\Gamma_1\widetilde{\bowtie} \Gamma_2$ and obtain a sufficient and necessary condition such that the spectrum of $\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma_1\widetilde{\bowtie}\Gamma_2$ are symmetric, from which we obtain more general results on maximum degree of the induced subgraphs.