10182

# Graceful Graphs

A graceful graph has edges labeled 1 to , with each edge label equal to the absolute difference between the labels of its vertices. (Assume one vertex is labeled 0 and no two vertex labels are the same.)
Two of the joined vertices have labels 0 and . The upper-right and lower-left squares of the adjacency matrix thus always contain a 1, shown as a black square here. Each diagonal parallel to the main diagonal of must have exactly one black square for the graph to be graceful. There are two choices for the edge labeled ; it connects either 0 and or 1 and . There are three choices for the next edge, labeled . Thus, there are exactly graceful graphs with edges.
There are three nongraceful graphs with 5 vertices: , , and the butterfly graph. All tree graphs with up to 35 vertices are graceful.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.