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

Recently, Huang showed that every ‐vertex induced subgraph of the ‐dimensional hypercube has maximum degree at least . In this paper, we discuss the induced subgraphs of Cartesian product graphs and semistrong product graphs to generalize Huang's result. Let be a connected signed bipartite graph of order and be a connected signed graph of order . By defining two kinds of signed product of and , denoted by and , we show that if and have exactly two distinct adjacency eigenvalues and , respectively, then every ‐vertex induced subgraph of (resp., ) has maximum degree at least (resp., ). Moreover, we discuss the eigenvalues of and and obtain a sufficient and necessary condition such that the spectrum of and is symmetric with respect to 0, from which we obtain more general results on maximum degree of the induced subgraphs.