On Extremal Values Of Total Structure Connectivity and Narumi-Katayama Indices on the Class of all Unicyclic and Bicyclic Graphs

‎The total structure connectivity and Narumi-Katayama indices of a simple graph $G$ are defined as $TS(G)={\prod_{{u}\in{V(G)}}}{\frac{1}{\sqrt {{d_{u}}}}}$ and $ NK(G)={\prod_{{u}\in{V(G)}}{{d_{u}}}}$ respectively‎, ‎where $d _{u} $ represents the degree of vertex $ u $ in $ G $‎. ‎In this paper‎, ‎we determine the extremal values of total structure connectivity index on the class of unicyclic and bicyclic graphs and characterize the corresponding extremal graphs‎. ‎In addition‎, ‎we determine the bicyclic graphs extremal with respect to the Narumi-Katayama index‎.