Rooted product of graphs - Wikipedia, the free encyclopedia

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The rooted product of graphs.

In mathematical graph theory, the rooted product of a graph G and a rooted graph H is defined as follows: take |V(G)| copies of H, and for every vertex v_i of G, identify v_i with the root node of the i-th copy of H.

More formally, assuming that V(G) = {g₁, ..., g_n}, V(H) = {h₁, ..., h_m} and that the root node of H is h₁, define

where

and

If G is also rooted at g₁, one can view the product itself as rooted, at (g₁, h₁). The rooted product is a subgraph of the cartesian product of the same two graphs.

The rooted product is especially relevant for trees, as the rooted product of two trees is another tree. For instance, Koh et al. (1980) used rooted products to find graceful numberings for a wide family of trees.

[edit] References

• Godsil, C. D.; McKay, B. D. (1978). "A new graph product and its spectrum". Bull. Austral. Math. Soc. 18 (1): 21–28. doi:10.1017/S0004972700007760. MR0494910. http://cs.anu.edu.au/~bdm/papers/RootedProduct.pdf. 

Koh, K. M.; Rogers, D. G.; Tan, T. (1980). "Products of graceful trees". Discrete Mathematics 31 (3): 279–292. doi:10.1016/0012-365X(80)90139-9. MR0584121.