King polyomino graph

Chart. 1. Nonadjacency relations. 0(G, k) is a general relation which counts the number of k nonadjacent objects in G. p(T, k) = the number of k-matchings in T, o(A, k) is the number of selections of k nonadjacent vertices in A, r(B, k) is the fcth resonant number of B, [Pg.279]

Fig. 5. A Gutman tree, T, an unbranched benzenoid system, B, a Clar graph, A, a king polyomino-graphy, P, and a rook board, Pr...

Fig. 6. All 3-matchings in T4(2,1,0,1) and the corresponding object-distribution in other equinumerical graphs, viz., an unbranched benzenoid hydrocarbon, a king polyomino and a rook board...

Number of ways of arranging k non-taking kings on a polyomino graph. 

Let us consider more closely the set of objects of Fig 7 associated with the same adjacency matrix A We will refer to these as the set T, A, B, P standing, respectively, for a caterpillar tree, a Clar graph (As L(T)) a benzenoid graph and a king polyomino The grai invariants that we will consider in each case are as follows ... 

The term "pseudo" king polyomino is used here to indicate that they ate not proper polyomino graphs [37] but contain factors of the variable x The same terminology might be extended to caterpillars. See, S- El-Basil, J- Math- Chem-, i, 161 (1987) D-H- Rouvray smd S- El-Basil, J. Molecular Structure (Theochem-), 165. 13 (1988). 

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... 

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