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.
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.
The characteristic polynomial of the butterfly graph is .
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- Ando, Kiyoshi (2007), "Contractible edges in a k-connected graph", Discrete geometry, combinatorics and graph theory, Lecture Notes in Comput. Sci., 4381, Springer, Berlin, pp. 10–20, doi:10.1007/978-3-540-70666-3_2, MR 2364744.