Nullity of a graph with a cut-edge. (English) Zbl 1461.05117
Summary: The nullity of a graph is known to be an analytical tool to predict reactivity and conductivity of molecular \(pi\)-systems. In this paper we consider the change in nullity when graphs with a cut-edge, and others derived from them, undergo geometrical operations. In particular, we consider the deletion of edges and vertices, the contraction of edges and the insertion of an edge at a coalescence vertex. We also derive three inequalities on the nullity of graphs along the same lines as the consequences of the Interlacing Theorem. These results shed light, in the tight-binding source and sink potential model, on the behaviour of molecular graphs which allow or bar conductivity in the cases when the connections are either distinct or ipso.
MSC:
05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
05C92 | Chemical graph theory |
92E10 | Molecular structure (graph-theoretic methods, methods of differential topology, etc.) |