An inequality between variable wiener index and variable szeged index

Abstract A well-known inequality between the Szeged and Wiener indices says that Sz ( G ) = ∑ e = i j ∈ E ( G ) n e ( i ) n e ( j ) ≥ ∑ { u , v } d ( u , v ) = W ( G ) for every graph G. In the past, variable variations of the standard topological indices were defined. Following this line, we study a natural generalisation of the above inequality, namely ∑ e = i j ∈ E ( G ) ( n e ( i ) n e ( j ) ) α ≥ ∑ { u , v } d ( u , v ) α . We show that for all trees the inequality is true if α > 1, and the opposite inequality holds if 0 ≤ α  n + 3 edges, but the opposite one does not. For general graphs we solve also the case α