Hosoya index

The Hosoya index of a graph is its total number of matchings, including the empty matching. It counts sets of pairwise vertex-disjoint edges.

Matching (graph theory): A set of edges in a graph with no shared endpoints. The Hosoya index counts every matching, from the empty set to the largest ones.
Matching polynomial: A graph polynomial whose coefficient for each degree counts matchings of that size. Evaluating its coefficients and summing them gives the Hosoya index.
Chemical graph theory: The study of chemical properties and phenomena using graph representations of molecules. The index became a molecular descriptor for comparing structures represented as graphs.
Independence number: The largest number of pairwise nonadjacent vertices in a graph. It records only the largest independent vertex set, while the Hosoya index counts all matchings.
Induced subgraph: A graph formed from a chosen vertex subset and all edges joining vertices in that subset. Deleting vertices leaves induced subgraphs whose indices support useful counting recurrences.
Path graph: A graph whose vertices can be arranged in a single path with edges only between consecutive vertices. Its matching counts follow a Fibonacci recurrence, giving a simple model for index computation.
Molecular descriptor: A numerical or symbolic property computed from a molecular structure for analysis or prediction. The Hosoya index is a topological descriptor derived from a molecule's connectivity graph.
Chromatic polynomial: A polynomial counting proper vertex colorings of a graph by the number of available colors. It counts colorings rather than edge-disjoint sets, despite also encoding graph structure in a polynomial.
Graph deletion: The operation of removing vertices or edges from a graph to form a smaller graph. Vertex deletion separates matchings according to whether a chosen vertex is unmatched or matched.
Tree (graph theory): A connected graph containing no cycles. Removing a leaf yields a compact recurrence for the total matching count.

Linked from 1 page