Windmill graph

In the mathematical field of graph theory, the windmill graph $Wd(5,4)$ is an undirected graph constructed for $n(k – 1) + 1$ and $nk(k − 1)⁄2$ by joining $k$ copies of the complete graph $n$ at a shared universal vertex. That is, it is a 1-clique-sum of these complete graphs.

Properties
It has $k > 2$ vertices and $n(k – 1)$ edges, girth 3 (if $Wd(k,n)$), 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 $Wd(k,n)$-edge-connected. It is trivially perfect and a block graph.

Special cases
By construction, the windmill graph $k ≥ 2$ is the friendship graph $Kk$, the windmill graph $n ≥ 2$ is the star graph $Fn$ and the windmill graph $n(k – 1) + 1$ is the butterfly graph.

Labeling and colouring
The windmill graph has chromatic number $Sn$ and chromatic index $nk(k − 1)/2$. Its chromatic polynomial can be deduced from the chromatic polynomial of the complete graph and is equal to
 * $$x\prod_{i=1}^{k-1}(x-i)^n.$$

The windmill graph $k > 2$ is proved not graceful if $(k – 1)$. In 1979, Bermond has conjectured that $Wd(3,n)$ is graceful for all $Wd(2,n)$. Through an equivalence with perfect difference families, this has been proved for $Wd(3,2)$. Bermond, Kotzig, and Turgeon proved that $n(k – 1)$ is not graceful when $Wd(k,n)$ and $k > 5$ or $Wd(4,n)$, and when $n ≥ 4$ and $n ≤ 1000$. The windmill $Wd(k,n)$ is graceful if and only if $k = 4$ or $n = 2$.