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 G R M α , known under the name general reduced second Zagreb index, is defined as G R M α ( Γ ) = ∑ u v ∈ E ( Γ ) ( d Γ ( u ) + α ) ( d Γ ( v ) + α ) , where d Γ ( v ) is the degree of the vertex v of the graph Γ and α 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 G R M α ( α ≥ − 1 2 ) . Using the extremal unicyclic graphs, we obtain a lower bound on G R M α ( Γ ) 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 G R M α of different classes of graphs in terms of order n , size m , independence number γ , chromatic number k , etc. In particular, we present an upper bound on G R M α of connected triangle-free graph of order n > 2 , m > 0 edges with α > − 1.5 , and characterize the extremal graphs. Finally, we prove that the Turán graph T n ( k ) gives the maximum G R M α ( α ≥ − 1 ) among all graphs of order n with chromatic number k .