complete graphK4 has the ten matchings shown, so its Hosoya index is ten, the maximum for any four-vertex graph.
The Hosoya index, also known as the Z index, of a
graph is the total number of
matchings in it. The Hosoya index is always at least one, because the
empty set of edges is counted as a matching for this purpose. Equivalently, the Hosoya index is the number of non-empty matchings plus one. The index is named after
Haruo Hosoya. It is used as a
topological index in
chemical graph theory.
In his article, "The Topological Index Z Before and After 1971," on the history of the notion and the associated inside stories, Hosoya writes that he introduced the Z index to report a good correlation of the
boiling points of
alkaneisomers and their Z indices, basing on his unpublished 1957 work carried out while he was an undergraduate student at the
University of Tokyo.
alkane, for the purposes of the Hosoya index, may be represented as a
path graph without any branching. A path with one vertex and no edges (corresponding to the
methane molecule) has one (empty) matching, so its Hosoya index is one; a path with one edge (
ethane) has two matchings (one with zero edges and one with one edges), so its Hosoya index is two.
Propane (a length-two path) has three matchings: either of its edges, or the empty matching. n-
butane (a length-three path) has five matchings, distinguishing it from
isobutane which has four. More generally, a matching in a path with edges either forms a matching in the first edges, or it forms a matching in the first edges together with the final edge of the path. This case analysis shows that the Hosoya indices of linear alkanes obey the recurrence governing the
Fibonacci numbers, and because they also have the same base case they must equal the Fibonacci numbers. The structure of the matchings in these graphs may be visualized using a
The largest possible value of the Hosoya index, on a graph with vertices, is given by the
complete graph. The Hosoya indices for the complete graphs are the
1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (sequence A000085 in the
^Hosoya, Haruo (1971), "Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons", Bulletin of the Chemical Society of Japan, 44 (9): 2332–2339,
^Gutman, Ivan (1991), "Polynomials in graph theory", in Bonchev, D.; Rouvray, D. H. (eds.), Chemical Graph Theory: Introduction and Fundamentals, Mathematical Chemistry, vol. 1, Taylor & Francis, pp. 133–176,