Abstract
We suggest the notion of H-centred surface area of a graph G, where H is a subgraph of G, i.e. the number of vertices in G at a certain distance from H, and focus on the special case when H is a length two path to derive an explicit formula for the length two path-centred surface areas of the general and scalable arrangement graph.
Acknowledgements
We greatly appreciate the thorough review and insightful comments made by two anonymous referees, which led to a clarification and further improvement of this paper.
Notes
1. A cycle is trivial if it contains exactly one symbol, called a fixed point. It is non-trivial otherwise.
2. When 1 occurs in an external cycle, we need to consider an additional symbol, i.e. the external symbol associated with the external cycle where 1 occurs.
3. We notice that , which swaps two internal symbols with one external symbol in An, k, k∈[2, n−1], is the only viable length two path allowed for An, n−1 ≡ Sn.