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 (Formula presented.), known under the name general reduced second Zagreb index, is defined as (Formula presented.), where (Formula presented.) is the degree of the vertex v of the graph (Formula presented.) and (Formula presented.) 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 (Formula presented.) (Formula presented.). Using the extremal unicyclic graphs, we obtain a lower bound on (Formula presented.) 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 (Formula presented.) of different classes of graphs in terms of order n, size m, independence number (Formula presented.), chromatic number k, etc. In particular, we present an upper bound on (Formula presented.) of connected triangle-free graph of order (Formula presented.), (Formula presented.) edges with (Formula presented.), and characterize the extremal graphs. Finally, we prove that the Turán graph (Formula presented.) gives the maximum (Formula presented.) among all graphs of order n with chromatic number k.