Windmill graph

In the mathematical field of graph theory, the windmill graph is an undirected graph constructed for and by joining copies of the complete graph at a shared universal vertex. That is, it is a 1-clique-sum of these complete graphs.

Properties

It has vertices and edges, girth 3, radius 1 and diameter 2.
It has vertex connectivity 1 because its central vertex is an articulation point; however, like the complete graphs from which it is formed, it is -edge-connected. It is trivially perfect and a block graph.

By construction, the windmill graph is the friendship graph, the windmill graph is the star graph and the windmill graph is the butterfly graph.

The windmill graph has chromatic number and chromatic index. Its chromatic polynomial can be deduced from the chromatic polynomial of the complete graph and is equal to
The windmill graph is proved not graceful if. In 1979, Bermond has conjectured that is graceful for all. Through an equivalence with perfect difference families, this has been proved for.
Bermond, Kotzig, and Turgeon proved that is not graceful when and or, and when and. The windmill is graceful if and only if or.