In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, the windmill graph is an

undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...

constructed for and by joining copies of the

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is c ...

at a shared

universal vertex In graph theory, a universal vertex is a vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a dominating vertex, as it forms a one-element dominating set in the graph. (It is not to be confused ...

. That is, it is a 1-clique-sum of these complete graphs.

Properties

It has vertices and edges, girth 3 (if ), radius 1 and diameter 2. It has

vertex connectivity Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

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 In graph theory, a branch of combinatorial mathematics, a block graph or clique tree. is a type of undirected graph in which every biconnected component (block) is a clique. Block graphs are sometimes erroneously called Husimi trees (after K� ...

Special cases

By construction, the windmill graph is the

friendship graph In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or -fan) is a planar, undirected graph with vertices and edges. The friendship graph can be constructed by joining copies of the cycle graph with a ...

, the windmill graph is the star graph and the windmill graph is the

butterfly graph In the mathematical field of graph theory, the butterfly graph (also called the bowtie graph and the hourglass graph) is a planar, undirected graph with 5 vertices and 6 edges. It can be constructed by joining 2 copies of the cycle graph with a ...

Labeling and colouring

The windmill graph has

chromatic number In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices o ...

and

chromatic index In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue ...

. Its

chromatic polynomial The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study ...

can be deduced form the chromatic polynomial of the complete graph and is equal to :

x\prod_^(x-i)^n.

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 .

Gallery

References

{{reflist Parametric families of graphs Perfect graphs