On vertex-degree-based molecular structure descriptors

Abstract

If \(G=(\mathbf{V} \ \mathbf{E})\) is a molecular graph and \(d_u\) is the degree of vertex \(u\), then the first and second Zagreb indices are \(\sum_{u\in \mathbf{V}} d_u^2\) and \(\sum_{uv\in \mathbf{E}} _u\,d_v\),respectively. These molecular structure descriptors, introduced in the 1970s, have been much studied. Yet, a number of their properties that seem to have evaded attention so far are established in this work for the first time. Also a more recent degree-based descriptor, the geometric-arithmetic index, equal to \ \sum_{uv\in\mathbf{E}} \sqrt{d_u\,d_v}/[(d_u + d_v)/2]\), is analyzed. It is demonstrated that instead of the ratio of geometric and arithmetic means, almost any other means could be used.