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The general sum-connectivity index of a graph G is defined as $$\chi _\alpha (G)=\sum _{uv\in E(G)}(d(u)+d(v))^{\alpha }$$, where d(u) denotes the degree of a vertex u in G and $$\alpha $$ is a real number. In this paper, we determine the minimum general sum-connectivity indices of trees with n vertices and matching number m, where $$n=2m$$ for $$\alpha \le -\,2$$ and $$2m\le n\le 3m+1$$ for $$\alpha >1$$, respectively. The corresponding extremal graphs are also characterized.

The general sum-connectivity index of a graph G is defined as (G)=uvE(G)(d(u)+d(v)), where d(u) denotes the degree of a vertex u in G and is a real number. In this paper, we present the maximum general sum-connectivity indices of trees and chemical trees with given number of pendent vertices for 1

The atom-bond connectivity ( A B C ) index of a graph G is defined as the sum of the weights d ( u ) + d ( v ) − 2 d ( u ) d ( v ) of all edges u v of G , where d ( u ) denotes the degree of a vertex u in G . In this short note, we determine the maximum A B C index for graphs with edge-connectivity one and characterize the corresponding extremal graphs. This answers a recent question proposed by Zhang et al. (2016).

The atom-bond connectivity index ( A B C ) and the first geometric-arithmetic index ( G A ) are two well-known molecular descriptors, which are found to be useful tools in QSPR/QSAR investigations. In this work, we obtain a relation between these two indices for simple connected graphs on n ? 3 vertices with minimum degree at least s and maximum degree at most t , where 1 ? s ? t ? n - 1 and t ? 2 . Using this relation, we prove that if t ? 4 s 2 - 3 s + 1 , then the A B C index is always less than the G A index and this bound is best possible for s ? 2 .

The atom-bond connectivity (ABC) index is a much-studied molecular structure descriptor, based on the degrees of the vertices of the molecular graph. Recently, another vertex-degree-based topological index - the harmonic index (H) - attracted attention and gained popularity. We show how ABC and H are related.