The Hosoya polynomial decomposition for catacondensed benzenoid graphs Original Research Article

For a graph image with the vertex set image, we denote by image the distance between vertices image and image in image, by image the degree of vertex image. The Hosoya polynomial of image is image. The partial Hosoya polynomials of image are image for positive integer numbers image and image. It is shown that image and image for arbitrary catacondensed benzenoid graphs image and image with equal number of hexagons. As an application, we give an affine relationship between image with two other distance-based polynomials constructed by Gutman [I. Gutman, Some relations between distance-based polynomials of trees, Bulletin de l’Académie Serbe des Sciences et des Arts (Cl. Math. Natur.) 131 (2005) 1–7].