On the determinant of bipartite graphs

The nullity   of a graph GG, denoted by η(G)η(G), is the multiplicity of 0 in the spectrum of GG. 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, detA(G)=0detA(G)=0 if and only if η(G)>0η(G)>0. So, examining the determinant of a graph is a way to attack this problem. For a graph GG, we define the matching core   of GG to be the graph obtained from GG by successively deleting each pendant vertex along with its neighbour. In this paper, we show that the determinant of a graph GG with all cycle lengths divisible by four (e.g., the 1-subdivision of a bipartite graph), is 0 or (−1)|V(G)|/2(−1)|V(G)|/2. Furthermore, the determinant is 0 if and only if the matching core of GG is nonempty.