A palindromicity criterion for the h -polynomials of bipartite edge rings

We study a symmetry problem for the [Formula: see text]-polynomials of edge rings of bipartite graphs. Let [Formula: see text] be a bipartite graph and write [Formula: see text]. We prove that if [Formula: see text] is pseudo-Gorenstein and [Formula: see text], then [Formula: see text] is Gorenstein. Equivalently, under these assumptions the [Formula: see text]-polynomial of [Formula: see text] is palindromic. The proof treats the [Formula: see text]-connected case first by translating the numerical condition [Formula: see text] into a tight-separation condition for non-edges, and then passes to arbitrary bipartite graphs using the block decomposition. We also construct a blockwise minimal Gorenstein closure, obtained by adjoining all non-edges not separated by tight acceptable sets, and show that this construction preserves the next-to-leading coefficient of the [Formula: see text]-polynomial.