The PI Index of polyomino chains

The PI index is a graph invariant defined as the summation of the sums of neu(e|G)neu(e|G) and nev(e|G)nev(e|G) over all the edges e=uve=uv of a connected graph GG, i.e., PI(G)=∑e∈E(G)[neu(e|G)+nev(e|G)], where neu(e|G)neu(e|G) is the number of edges of GG lying closer to uu than to vv and nev(e|G)nev(e|G) is the number of edges of GG lying closer to vv than to uu. An efficient formula for calculating the PI index of polyomino chains is given, and the bounds for the PI index of polyomino chains are established.