On General Reduced Second Zagreb Index of Graphs

Graph-based molecular structure descriptors (often called topological indices) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant GRM(alpha), known under the name general reduced second Zagreb index, is defined as GRM(alpha) (Gamma)=Sigma(uv is an element of E(Gamma))(d(Gamma)(u) + alpha)(d(Gamma)(v) + alpha), where d(Gamma)(v) is the degree of the vertex v of the graph Gamma and alpha is any real number. In this paper, among all trees of order n, and all unicyclic graphs of order n with girth g, we characterize the extremal graphs with respect to GRM(alpha) (alpha >= -1/2). Using the extremal unicyclic graphs, we obtain a lower bound on GRM(alpha) (Gamma) of graphs in terms of order n with k cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on GRM(alpha) of different classes of graphs in terms of order n, size m, independence number gamma, chromatic number k, etc. In particular, we present an upper bound on GRM(alpha) of connected triangle-free graph of order n > 2, m > 0 edges with alpha >-1.5, and characterize the extremal graphs. Finally, we prove that the Turan graph T-n(k) gives the maximum GRM(alpha)(alpha >=-1) among all graphs of order n with chromatic number k.

Automated summary

Graph-based molecular structure descriptors, known as topological indices, are important for modeling the properties of molecules and designing compounds. In this paper, the graph invariant GRM(alpha) is studied, and extremal graphs with respect to GRM(alpha) are characterized.