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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 $Γ_1$ be a connected signed bipartite graph of order $n$ and $Γ_2$ be a connected signed graph of order $m$. By defining two kinds of signed product of $Γ_1$ and $Γ_2$, denoted by $Γ_1\widetilde{\Box}Γ_2$ and $Γ_1\widetilde{\bowtie} Γ_2$, we show that if $Γ_1$ and $Γ_2$ have exactly two distinct adjacency eigenvalues $\pmθ_1$ and $\pmθ_2$ respectively, then every $(\frac{1}{2}mn+1)$-vertex induced subgraph of $Γ_1\widetilde{\Box}Γ_2$ (resp. $Γ_1\widetilde{\bowtie} Γ_2$) has maximum degree at least $\sqrt{θ_1^2+θ_2^2}$ (resp. $\sqrt{(θ_1^2+1)θ_2^2}$). Moreover, we discuss the eigenvalues of $Γ_1\widetilde{\Box} Γ_2$ and $Γ_1\widetilde{\bowtie} Γ_2$ and obtain a sufficient and necessary condition such that the spectrum of $Γ_1\widetilde{\Box}Γ_2$ and $Γ_1\widetilde{\bowtie}Γ_2$ are symmetric, from which we obtain more general results on maximum degree of the induced subgraphs.