The inertia of unicyclic graphs and the implications for closed-shells

作者：Daugherty Sean^*

来源：Linear Algebra and Its Applications, 2008, 429(4): 849-858.

DOI：10.1016/j.laa.2008.04.013

• eigenvalue problems
• molecular structures
• graph algorithms
• matchings
• unicyclic graphs
• closed-shell
• molecular stability

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.

• 出版日期2008-8-1