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
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]