On the determinant of bipartite graphs

The nullity of a graph G , denoted by η ( G ) , is the multiplicity of 0 in the spectrum of G . Nullity of a (molecular) graph (e.g., a bipartite graph corresponding to an alternant hydrocarbon) has important applications in quantum chemistry and Hückel molecular orbital (HMO) theory. A famous problem, posed by Collatz and Sinogowitz in 1957, asks to characterize all graphs with positive nullity. Clearly, det A ( G ) = 0 if and only if η ( G ) > 0 . So, examining the determinant of a graph is a way to attack this problem. For a graph G , we define the matching core of G to be the graph obtained from G by successively deleting each pendant vertex along with its neighbour. In this paper, we show that the determinant of a graph G with all cycle lengths divisible by four (e.g., the 1-subdivision of a bipartite graph), is 0 or ( − 1 ) | V ( G ) | / 2 . Furthermore, the determinant is 0 if and only if the matching core of G is nonempty.