Butterfly graph
| Butterfly graph | |
|---|---|
 |
|
| Vertices | 5 |
| Edges | 6 |
| Radius | 1 |
| Diameter | 2 |
| Girth | 3 |
| Automorphisms | 8 (D4) |
| Chromatic number | 3 |
| Chromatic index | 4 |
| Properties |
Planar Unit distance Eulerian Not graceful |
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 C3 with a common vertex and is therefore isomorphic to the friendship graph F2.
The butterfly graph has diameter 2 and girth 3, radius 1, chromatic number 3, chromatic index 4 and is both Eulerian and unit distance. It is also a 1-vertex-connected graph and a 2-edge-connected graph.
There are only 3 non-graceful simple graphs with five vertices. One of them is the butterfly graph. The two others are cycle graph C5 and the complete graph K5.
A graph is bowtie-free if it has no butterfly as an induced subgraph. The triangle-free graphs are bowtie-free graphs, since every butterfly contains a triangle.
In a k-vertex-connected graph, and edge is said k-contractible if the contraction of the edge results in a k-connected graph. Ando, Kaneko, Kawarabayashi and Yoshimoto proved that every k-vertex-connected bowtie-free graph has a k-contractible edge.
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