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

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

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 ${\displaystyle -(x-1)(x+1)^{2}(x^{2}-x-4)}$.

References