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.^[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.

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

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