Equivalent graphs, mathematical objects

First we will focus attention on selected topics relating to the equivalence between benzenoid hydrocarbons, and special types of graphs and other mathematical objects that we can associate with benzenoids- In particular we will explore relations involving caterpillar trees [3] associated with catacondensed benzenoids and their line graphs [17] called, as already mentioned, Clar graphs [4]. Also relations involving "boards" (known technically as polyominos) of special properties such as those associated with "king" and "rook" pieces of chess... 

Common conventions of graph theory are followed here (see, e.g.. Refs. 8-10), but a rigid distinction is not always maintained between chemical and mathematical contexts. So the term graph is used interchangeably in both its strict sense as a mathematical object and as a shorthand term for its realization as an actual (usually carbon) molecule. Some equivalent pairs of terms are treated as being synonymous atom = vertex, bond = edge, valency = degree, and so on. 

The enumeration of Ld-sequences is equivalent to the enumeration of Gutman trees, which in the mathematical literature are called caterpillar trees (or caterpillars). These objects correspond to special trees in the graph-theoretical sense. 

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