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 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 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 C5 and the complete graph K5.
| 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 |
| Table of graphs and parameters | |
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][2] 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 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 C5 and the complete graph K5.[3]
Bowtie-free graphs
[edit]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.[4]
Algebraic properties
[edit]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 .
References
[edit]- ^ Weisstein, Eric W. "Butterfly Graph". MathWorld.
- ^ ISGCI: Information System on Graph Classes and their Inclusions. "List of Small Graphs".
- ^ Weisstein, Eric W. "Graceful graph". MathWorld.
- ^ Ando, Kiyoshi (2007), "Contractible edges in a k-connected graph", Discrete geometry, combinatorics and graph theory, Lecture Notes in Comput. Sci., vol. 4381, Springer, Berlin, pp. 10–20, doi:10.1007/978-3-540-70666-3_2, MR 2364744.