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

Recently, Huang showed that every ( 2 n − 1 + ) -vertex induced subgraph of the n-dimensional hypercube has maximum degree at least . In this paper, we discuss subgraphs Cartesian product graphs and semistrong to generalize Huang's result. Let Γ be a connected signed bipartite graph order m. By defining two kinds , denoted by □ ˜ ⋈ show if have exactly distinct adjacency eigenvalues ± θ respectively, then m (resp., ). Moreover, obtain sufficient necessary condition such spectrum is symmetric with respect 0, from which more general results on subgraphs.