The Merrifield–Simmons Indices and Hosoya Indices of Trees with k Pendant Vertices

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

“…This theorem was proven by Pan et al [84] and independently by Yu and Lv [117], and extended by Wang et al [104] for the Merrifield-Simmons index (they also provide the trees with second-largest Merrifield-Simmons index) as well as Lv et al [68] (secondlargest Merrifield-Simmons index and second-smallest Hosoya index). The problem seems to be much more complicated if one asks for the minimum Merrifield-Simmons index or maximum Hosoya index; Yan and Ye [110] (for the Hosoya index) and Deng and Guo [13] (for the Merrifield-Simmons index) provide partial (yet interesting) results in this direction for trees with many leaves (trees with n vertices and at least n 2 + 1 leaves).…”

“…As in the case of the previous theorem, some variants and related results exist, see for instance [68,102,112,117]. In particular, the following result was repeatedly used by Deng et al [8,10,11] in their study of (n, n + 1)-graphs (for convenience, we use a slightly different notation):…”

“…This is another instance of a class of graphs for which the extremal graphs for the two indices coincide: [84,117]) Among all trees with n vertices and k leaves (also known as pendant vertices), the tree that maximizes the Merrifield-Simmons index and minimizes the Hosoya index is the broom, i.e., the tree that results from identifying the center of a star S k with one end of a path of length n − k, see Fig. 5.…”

“…In the literature, this theorem appears in various variants and special cases (for example in the case that H 1 and H 2 are stars [9,12,67,68,116,117]). See also [58,60,97,112] for applications of this "transfer theorem", which is usually used to rule out specific substructures in extremal graphs.…”

Merrifield-Simmons and Hosoya index of thorn-regular graphs

“…Details on their theory as well as additional references can be found in the survey [16]. In particular, see [10] and [19] for trees, [17] for trees with given number of pendent vertices, [9] for trees with a given diameter, [4] for trees with bounded degree, [12] for unicyclic graphs, [8] for the unicyclic graphs with a given diameter, and [6] for unicyclic graphs with given number of cut vertices.…”