Some extremal properties of the multiplicatively weighted Harary index of a graph

Let $$G=(V_G, E_G)$$ G = ( V G , E G ) be a simple connected graph. The multiplicatively weighted Harary index of $$G$$ G is defined as $$H_M(G)=\sum _{\{u,v\}\subseteq V_G}\delta _G(u)\delta _G(v)\frac{1}{d_G(u,v)},$$ H M ( G ) = ∑ { u , v } ⊆ V G δ G ( u ) δ G ( v ) 1 d G ( u , v ) , where $$\delta _G(u)$$ δ G ( u ) is the vertex degree of $$u$$ u and $$d_G(u,v)$$ d G ( u , v ) is the distance between $$u$$ u and $$v$$ v in $$G.$$ G . This novel invariant is in fact the modification of the Harary index in which the contributions of vertex pairs are weighted by the product of their degrees. Deng et al. (J Comb Optim 2014, doi: 10.1007/s10878-013-9698-5 ) determined the extremal values on $$H_M$$ H M of graphs among $$n$$ n -vertex trees (resp. unicyclic graphs). In this paper, as a continuance of it, the monotonicity of $$H_M(G)$$ H M ( G ) under some graph transformations were studied. Using these nice mathematical properties, the extremal graphs among $$n$$ n -vertex trees with given graphic parameters, such as pendants, matching number, domination number, diameter, vertex bipartition, et al. are characterized, respectively. Some sharp upper bounds on the multiplicatively weighted Harary index of trees with given parameters are determined.