Diamond-free graphs and forbidden minor
A graph is diamond-free if it has no diamond as an induced subgraph. The triangle-free graphs are diamond-free graphs, since every diamond contains a triangle. The diamond-free graphs are locally clustered: that is, they are the graphs in which every neighborhood is a cluster graph.
The family of graphs in which each connected component is a cactus graph is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor. This minor is the diamond graph.
The characteristic polynomial of the diamond graph is . It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
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