TOPICS

# Graph Square

The square of a graph is defined as its second graph power.

The square of any biconnected graph is Hamiltonian (Fleischner 1974, Skiena 1990, p. 231). Mukhopadhyay (1967) has considered "square root graphs," whose square gives a given graph (Skiena 1990, p. 253).

Since raising any graph to the power of its graph diameter gives a complete graph, the square of any graph on nodes with graph diameter is a complete graph . Classes of such graphs include cocktail party graphs, complete graphs, complete bipartite graphs, complete tripartite graphs, dipyramid graphs, star graphs, and wheel graphs.

The following table summarizes the squares of some indexed families of graphs.

Graph Cube, Graph Power, Graph Product

## Explore with Wolfram|Alpha

More things to try:

## References

Fleischner, H. "The Square of Every Two-Connected Graph Is Hamiltonian." J. Combin. Th. Ser. B 16, 29-34, 1974.Mukhopadhyay, A. "The Square Root of a Graph." J. Combin. Th. 2, 290-295, 1967.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.

## Cite this as:

Weisstein, Eric W. "Graph Square." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GraphSquare.html