A Polyomino is an edge–conected union of cells in the planar square lattice. Polyominoes are very popular in mathematical recreations, and have found interest among mathematicians, physicists, biologists, and computer scientists as well. Because the chemical constitution of a molecule is conventionally represented by a molecular graph or network, the polyominoes have deserved the attention of the Organic Chemistry community. So, several molecular structure descriptors based in network structur...
A Polyomino is an edge–conected union of cells in the planar square lattice. Polyominoes are very popular in mathematical recreations, and have found interest among mathematicians, physicists, biologists, and computer scientists as well. Because the chemical constitution of a molecule is conventionally represented by a molecular graph or network, the polyominoes have deserved the attention of the Organic Chemistry community. So, several molecular structure descriptors based in network structural descriptors, have been introduced.
In particular, in the last decade a great amount of works devoted to calculate the Kirchhoff Index of linear polyominoes–like networks, have been published. In this work we deal with this class of polyominoes, that we call generalized linear polyominoes, that besides the
most popular class of linear polyomino chains, also includes cycles, Phenylenes and Hexagonal chains to name only a few. Because the Kirchhoff Index is the trace of the Green function of the network, here we obtain the Green function of generalized linear Polyominoes.
To do this, we understand a Polyomino as a perturbation of a path by adding weighted edges between opposite vertices. Therefore, unlike the techniques used by Yang and Zhang in 2008, that are based on the decomposition of the combinatorial Laplacian in structured blocks, here we obtain the Green function of a linear Polyomino from a perturbation of the combinatorial Laplacian. This approach deeply link linear Polyomino Green functions with the inverse M –matrix problem and specially, with the so–called Green matrices, of Gantmacher and Krein (1941).