Nullity of Graphs: an Updated Survey

Ivan Gutman, Bojana Borovićanin

The nullity $\eta=\eta(G)$ of a graph $G$ is the multiplicity of the number zero in the spectrum of $G$\,. The chemical importance of this graph-spectrum based invariant lies in the fact, that within the Hückel molecular orbital model, if $\eta(G)>0$ for the molecular graph $G$\,, then the corresponding chemical compound is highly reactive and unstable, or nonexistent. This chapter in an updated version of the an earlier survey [B. Borovićanin, I. Gutman, \emph{Nullity of graphs}, in: D. Cvetković, I. Gutman, Eds. \emph{Applications of Graph Spectra}, Math. Inst., Belgrade, 2009, pp. 107-122] and outlines both the chemically relevant aspects of $\eta$ (most of which were obtained in the 1970s and 1980s) and the general mathematical results on $\eta$ obtained recently.