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 n. In this paper, we discuss the induced subgraphs of Cartesian product graphs and semistrong 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 □ ˜ Γ 2 and Γ 1 ⋈ ˜ Γ 2, we show that if Γ 1 and Γ 2 have exactly two distinct adjacency eigenvalues ± θ 1 and ± θ 2, respectively, then every ( 1 2 m n + 1 )‐vertex induced subgraph of Γ 1 □ ˜ Γ 2 (resp., Γ 1 ⋈ ˜ Γ 2) has maximum degree at least θ 1 2 + θ 2 2 (resp., ( θ 1 2 + 1 ) θ 2 2). Moreover, we discuss the eigenvalues of Γ 1 □ ˜ Γ 2 and Γ 1 ⋈ ˜ Γ 2 and obtain a sufficient and necessary condition such that the spectrum of Γ 1 □ ˜ Γ 2 and Γ 1 ⋈ ˜ Γ 2 is symmetric with respect to 0, from which we obtain more general results on maximum degree of the induced subgraphs.