Overconstrained mechanism

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 unit distance. It is also a 1-vertex-connected graph and a 2-edge-connected graph.

There are only 3 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.

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, and edge is said 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.

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

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.