the padmakar-ivan (pi) index is a wiener-szeged-like topological index which reflectscertain structural features of organic molecules. the pi index of a graph g is the sum of alledges uv of g of the number of edges which are not equidistant from the vertices u and v. inthis paper we obtain the second and third extremals of catacondensed hexagonal systems withrespect to the pi index.

The vertex set of the resonance graph of a hexagonal graph G consists of 1-factors of G, two 1-factors being adjacent whenever their symmetric difference forms the edge set of a hexagon of G. A decomposition theorem for the resonance graphs of catacondensed hexagonal graph is proved. The theorem intrinsically uses the Cartesian product of graphs. A canonical binary coding of 1-factors of cataco...

In this paper the Clar covering polynomial of a hexagonal system is introduced. In fact it is a kind of F polynomial [4] of a graph, and can be calculated by recurrence relations. We show that the number of aromatic sextets (in a Clar formula), the number of Clar formulas, the number of Kekule structures and the first Herndon number for any Kekulean hexagonal system can be easily obtained by it...

Hexagonal chains are a special class of catacondensed benzenoid system and phenylene polycyclic aromatic compounds. Recently, A family Sombor indices was introduced by Gutman in the chemical graph theory. It had been examined that these may be successfully applied on modeling thermodynamic properties In this paper, we study expected values random hexagonal chains, consider some graphs such as g...

It is proved that a bipartite 2-connected plane graph in which the common boundary of adjacent faces is a simple curve is 1-cycle resonant if and only if the outer face of G is alternating and each inner vertex has degree two. This extends a result from [X. Guo, F. Zhang, k-cycle resonant graphs, Discrete Math. 135 (1994) 113-120] that a hexagonal system is 1-cycle resonant if and only if it is...

It is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two ...

Lei, Yeh and Zhang put forward the anti-forcing number af(G,M) for a perfect matching M in graph G, which is minimum of edges G not whose deletion results subgraph with unique M. The numbers all matchings form spectrum G. polynomial Af(G,x) counting classifying possessing same In this paper, we deduce recurrence formula continuity catacondensed hexagonal systems.

Let H be a hexagonal system. The Z-transformation graph Z(H) is the graph where the vertices are the perfect matchings of H and where two perfect matchings are joined by an edge provided their symmetric difference is a hexagon of H (Z. Fu-ji et al., 1988). In this paper we prove that Z(H) has a Hamilton path if H is a catacondensed hexagonal system. A hexagonal system [ll], also called honeycom...

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. The PI index of a graph G is the sum of all edges uv of G of the number of edges which are not equidistant from the vertices u and v. In this paper we obtain the second and third extremals of catacondensed hexagonal systems with respect to the PI index.