On the intersection rank of a graph (Q1198514)

Let \(G=(V,E)\) be a simple graph. Let \(\Omega_ V\) and \(\Omega_ E\) be free groups over \(V\) and \(E\) with coefficients from GF(2). \(\Omega_ V\) and \(\Omega_ E\) can be considered as binary vector spaces. A boundary operator \(\vartheta:\Omega_ E\to\Omega_ V\) and a coboundary operator \(\delta:\Omega_ V\to\Omega_ E\) are defined (they are linear maps). The cycle group \(\Gamma\) is defined to be the kernel of \(\vartheta\), and the coboundary group \(\Delta\) is the image of \(\vartheta\). The dimension of \(\Gamma\cap\Delta\) is called the intersection rank of \(G\). Basic equations on the intersection rank are derived. Special classes of graphs considered in the paper are bipartite graphs and graphs obtained from \(t\)-designs and circulant matrices.