Fast computation of the Wiener index of fasciagraphs and rotagraphs

The notion of a polygraph was introduced in chemical graph theory as a formalization of the chemical notion of polymers.’ Fasciagraphs and rotagraphs form an important class of polygraphs. In the language of graph theory they describe polymers with open ends and polymers that are closed upon themselves, respectively. They are highly structured, and this structure makes it possible to design efficient algorithms for computing several graph invariants.2 In this paper we show how the structure of fasciagraphs and rotagraphs can be used to obtain efficient algorithms for computing the Wiener index of such graphs. More precisely, if we regard basic arithmetic operations such as addition and multiplication to take a constant time, then the time complexity of our improved algorithms (theorem 5 ) depends only on the size k of a monograph in the polygraph and is independent of the number of monographs n. The paper is organized as follows. Motivation for studying such problems and definitions of polygraphs, rotagraphs, and fasciagraphs are given in section 1. Section 2 describes matrix approach to the computation of the Wiener index of fasciagraphs and rotagraphs. Two basic algorithms that realize this approach are presented (algorithms A and B). In section 3 possible extensions of these algorithms are briefly sketched. Using more sophisticated mathematical methods this approach is further extended, and the two algorithms