| Butterfly graph | |
| Vertices: | 5 |
| Edges: | 6 |
| Automorphisms: | 8 |
| Diameter: | 2 |
| Radius: | 1 |
| Girth: | 3 |
| 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.[1] It can be constructed by joining 2 copies of the cycle graph with a common vertex and is therefore isomorphic to the friendship graph .
The butterfly graph has diameter 2 and girth 3, radius 1, chromatic number 3, chromatic index 4 and is both Eulerian and a penny graph (this implies that it is unit distance and planar). It is also a 1-vertex-connected graph and a 2-edge-connected graph.
There are only three non-graceful simple graphs with five vertices. One of them is the butterfly graph. The two others are cycle graph and the complete graph .
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, an edge is said to be 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.[2]
The full automorphism group of the butterfly graph is a group of order 8 isomorphic to the dihedral group D4, the group of symmetries of a square, including both rotations and reflections.
The characteristic polynomial of the butterfly graph is
-(x-1)(x+1)2(x2-x-4)