The Hosoya index and the Merrifield–Simmons index

In this article, we give sharp bounds on the Hosoya index and the Merrifield–Simmons index for connected graphs of fixed size. As a consequence, we determine all connected graphs of any fixed order and size which maximize the Merrifield–Simmons index. Sharp lower bounds on the Hosoya index are known for graphs of order n and size \(m\in [n-1,2n-3]\cup \left( {n-1\atopwithdelims ()2},{n\atopwithdelims ()2}\right] \); while sharp upper bounds were only known for graphs of order n and size \(m\le n+2\). We give sharp upper bounds on the Hosoya index for dense graphs with \(m\ge {n\atopwithdelims ()2}-2n/3\). Moreover, all extreme graphs are also determined.     

On the Randić index

The Randić index of an organic molecule whose molecular graph is G is defined as the sum of (d(u)d(v))^−1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Discrete Mathematics 257, 29–38 by Delorme et al. gave a best-possible lower bound on the Randić index of a triangle-free graph G with given minimum degree δ(G). In the paper, we first point out a mistake in the proof of their result (Theorem 2 of [2002]), and then we will show that the result holds when δ(G) ≥ 2.AMS subject classification: 05C18     

An improved molecular connectivity index

Through modification of the delta values of the molecular connectivity indexes, and connecting the quantum chemistry with topology method effectively, the molecular connectivity indexes are converted into quantum-topology indexes. The modified indexes not only keep all information obtained from the original molecular connectivity method but also have their own virtue in application, and at the same time make up some disadvantages of the quantum and molecular connectivity methods.     

Randić index and lexicographic order

Let T be a tree and consider the Randi index (T)= ), where v _i –v _j runs over all edges of T and (v _i ) denotes the degree of the vertex v _i . Using counting arguments we show that the Randi index, is monotone increasing over the well (lexicographic order) ordered sequence of trees with unique branched vertex.