Windmill graph

Windmill graph
Windmill graph
	The Windmill graph Wd(5,4).
Vertices	n(k – 1) + 1
Edges	⁠nk(k − 1)/2⁠
Radius	1
Diameter	2
Girth	3 if k > 2
Chromatic number	k
Chromatic index	n(k – 1)
Notation	Wd(k,n)
	Table of graphs and parameters

In the mathematical field of graph theory, the windmill graph $Wd(k, n)$ is an undirected graph constructed for $k \geq 2$ and $n \geq 2$ by joining $n$ copies of the complete graph $K k$ at a shared universal vertex. That is, it is a 1-clique-sum of these complete graphs.^[1]

^ Gallian, J. A. (3 January 2007). "A dynamic survey of graph labeling" (PDF). Electronic Journal of Combinatorics. DS6: 1–58. MR 1668059.

Windmill graph
The Windmill graph $Wd(5,4)$ .
Vertices	$n (k - 1) + 1$
Edges	$.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠nk(k − 1)/2⁠$
Radius	1
Diameter	2
Girth	3 if $k > 2$
Chromatic number	$k$
Chromatic index	$n (k - 1)$
Notation	$Wd(k, n)$
Table of graphs and parameters