The inertia of a graph is an integer triple specifying the number of negative, zero, and positive eigenvalues of the adjacency matrix of the graph. A unicyclic graph is a simple connected graph with an equal number of vertices and edges. This paper characterizes the inertia of a unicyclic graph in terms of maximum matchings and gives a linear-time algorithm for computing it. Chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is (properly) closed-shell, having exactly half of its eigenvalues greater than zero, because this designates a stable electron configuration. The inertia determines whether a graph is closed-shell, and hence the reported result gives a linear-time algorithm for determining this for unicyclic graphs. © 2008 Elsevier Inc. All rights reserved.
Daugherty, S. (2008). The inertia of unicyclic graphs and the implications for closed-shells. Linear Algebra and Its Applications, 429(4), 849–858. https://doi.org/10.1016/j.laa.2008.04.013