The Burge correspondence and crystal graphs

The Burge correspondence yields a bijection between simple labelled graphs and semistandard Young tableaux of threshold shape. We characterize the hook shape by peak valley conditions on arrays. This is first step towards an analogue Schensted’s result for RSK insertion which states that length longest increasing subword word largest row tableau under correspondence. Furthermore, we give crystal structure extremal vectors in this are precisely whose degree sequence hook-shaped.