Abstract
A large number of graph invariants of the form \( \sum_{uv} F(d_u,d_v) \) are studied in mathematical chemistry, where \( uv \) denotes the edge of the graph \( G \) connecting the vertices \( u \) and \( v \), and \( d_u \) is the degree of the vertex \( u \). Among them the variable Randić type lodeg index \( RLI_a \), with \( F(d_u,d_v)=\log\!^a d_u \log\!^a d_v \), for \( a>0 \), was found to have applicative properties. The aim of this paper is to obtain new inequalities for the variable Randić type lodeg index, and to characterize graphs extremal with respect to them. In particular, some of the open problems posed by Vukičević are solved in this paper; we characterize graphs with maximum and minimum values of the \( RLI_a \) index, for every \( a>0 \), in the following sets of graphs with \( n \) vertices: graphs, connected graphs, graphs with fixed minimum degree, connected graphs with fixed minimum degree, graphs with fixed maximum degree, and connected graphs with fixed maximum degree. Also, our results can be applied to a large class of topological indices of the form \( \sum_{uv \in E(G)} F(d_u,d_v) \), as variable sum lodeg index and variable inverse sum lodeg index, solving some of the open problems posed by Vukičević.
