Recently, Huang showed that every (2n−1+1)-vertex induced subgraph of the n-dimensional hypercube has maximum degree at least n
Nearby in the stack
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
Γ1
be a connected signed bipartite graph of order
n
and
Γ2
be a connected signed graph of order
m
. By defining two kinds of signed product of
Γ1
and
Γ2
, denoted by
Γ1□Γ2
and
Γ1⋈Γ2
, we show that if
Γ1
and
Γ2
have exactly two distinct adjacency eigenvalues
±θ1
and
±θ2
respectively, then every
(21mn+1)
-vertex induced subgraph of
Γ1□Γ2
(resp.
Γ1⋈Γ2
) has maximum degree at least
θ12+θ22
(resp.
(θ12+1)θ22
). Moreover, we discuss the eigenvalues of
Γ1□Γ2
and
Γ1⋈Γ2
and obtain a sufficient and necessary condition such that the spectrum of
Γ1□Γ2
and
Γ1⋈Γ2
are symmetric, from which we obtain more general results on maximum degree of the induced subgraphs.