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

Abstract

The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.Keywords Hosoya index · Merrifield-Simmons index · Extremal problem · Molecular graphs Mathematics Subject Classification (2000) 92E10 · 05C35 · 05C70 History and Chemical BackgroundIn 1971 the Japanese chemist Haruo Hosoya introduced a molecular-graph based structure descriptor [35], which he named topological index and denoted by Z. He showed that certain physico-chemical properties of alkanes (= saturated hydrocarbons)-in particular, their boiling points-are well correlated with Z. He defined the quantity Z in the following manner.Let G be a (molecular) graph. Denote by m(G, k) the number of ways in which k mutually independent edges can be selected in G. By definition, m(G, 0) = 1 for all graphs, and

Extremal Trees with respect to Number of(A,B,2C)-Edge Colourings

“…The numbers of the Fibonacci type play an important role in distinct areas of mathematics and they have many different applications and interpretations. Some of them are closely related to the Hosoya index ( ) (defined as a number of all matchings in the graph , including the empty matching) and the Merrifield-Simmons index ( ) (defined as a number of all independent sets in , including the empty set); see [2] and its references. It is well-known that (P ) = and (P ∘ K 1 ) = +1 , for ≥ 1, where P is an -vertex path, K is an -vertex complete graph, and ∘ denotes the corona of two graphs.…”

The Hosoya index and the Merrifield–Simmons index

“…More extreme results on these two indices for other class of graphs can be found in the survey [18]. In order to prove our theorems, we need a simple lemma which follows directly from the definitions.…”

Total Graph Interpretation of the Numbers of the Fibonacci Type

“…For other results related to the parameter NI( ) and their applications, see the last survey [7]. In this survey the authors collect and classify the results concerning the graph parameter NI( ), most of which are obtained quite recently.…”