Average distances in square-cell configurations

A square-cell configuration (“square animal”) is a subgraph of the square lattice in which all inner faces are 4-cycles. We determine explicit expressions for the sum (W) of the (topological) distances between all pairs of vertices of a square-cell configuration, as well as for the related average distance \documentclass{article}\pagestyle{empty}\begin{document}$\overline{W}$\end{document}. Such expressions are deduced for several families of symmetric square-cell configurations. For instance, if O(n) stands for the octagonal square-cell configuration with n circular levels, then W(O(n))=(211/5)n5−(181/3)n4+(109/3)n3−(35/3)n2+(22/15)n and W(O(n))=2[(7n2−10n+4)(7n2−10n+3)]−1W(O(n)). © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 611–617, 2000