Abstract
Let G be a graph. The Hosoya index of G , denoted by z G , is defined as the total number of its matchings. The computation of z G is NP-Complete. Wagner and Gutman pointed out that it is difficult to obtain results of the maximum Hosoya index among tree-like graphs with given diameter. In this paper, we focus on the problem, and a sharp bound of Hosoya indices of all bicyclic graphs with diameter of 3 is determined.
Highlights
Hosoya index is an important topological index introduced by Hosoya [1]
It was found that Hosoya index is related to a variety of physicochemical properties of alkanes
Another series of researches revealed the applicability of Hosoya index in the theory of conjugated π-electron systems [2, 3]
Summary
Hosoya index is an important topological index introduced by Hosoya [1]. It was found that Hosoya index is related to a variety of physicochemical properties of alkanes ( saturated hydrocarbons). The boiling points of alkanes are well correlated with Hosoya index. Wagner and Gutman [16] gave an exhaustive survey for Hosoya index, and they pointed out some open problems ( see [17]) as follows:. According to the open problems, Liu et al [17] discussed the problem in which unicyclic graph with diameter of 3 or 4 has the maximum Hosoya index. At is, which bicyclic graph with diameter of 3 has the maximum Hosoya index? Some upper bounds for Hosoya index of some special classes of bicyclic graphs with diameter of 3 are determined
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