Reciprocal Mean Square Index of Graphs

The reciprocal mean square (RMS) index of a graph G with edge set E(G) is defined by RMS(G)=∑uv∈E(G)[(d(u))2+(d(v))2]−1, where d(u) and d(v) are the degrees of vertices u and v, respectively. We first establish several bounds for the RMS index in terms of standard graph parameters (including the size, minimum degree, and maximum degree) and related degree-based indices, such as the forgotten index, the Sombor index, the reciprocal hyper-Zagreb index, the inverse degree index, and the harmonic index. We then determine the extremal values of the RMS index over the classes of n-order trees and unicyclic graphs for n≥3. For n-order trees, the path Pn and the star Sn are the extremal graphs for the considered index, as expected. However, for unicyclic graphs of sufficiently large order, the graph minimizing the RMS index is, rather surprisingly, not the one containing a universal vertex. This indicates that the extremal behavior of the RMS index may differ from that of many existing degree-based indices.