another approach to a conjecture about the exponential reduced sombor index of molecular trees

‎for a graph g‎, ‎the exponential reduced sombor index (ersi)‎, ‎denoted by esored , ‎is ∑‎uv∈e(g) e√(dg(v)-1)^2+(dg(u)-1)^2), ‎where dg(v) is the degree of vertex v‎. ‎the authors in [on the reduced sombor index and its applications‎, ‎match commun‎. ‎math‎. ‎comput‎. ‎chem‎. ‎86 (2021) 729–753] conjectured that for each molecular tree t of order n‎,  esored‎(t)≤(2/3) (n+1) e^3 +(1/3) (n5) e 3√2, where n≡2 (mod 3), esored‎(t)≤(1/3) (2n+1) e^3 +(1/3) (n13) e^3√2 + 3e√13 , where n≡1 (mod 3) and  esored‎(t)≤(2/3) ne^3 +(1/3) (n-9) e3√2 + 2e√10 , where n≡0 (mod 3). ‎recently‎, ‎hamza and ali [on a conjecture regarding the exponential reduced sombor index of chemical trees‎. ‎discrete math‎. ‎lett‎. ‎9 (2022) 107–110] proved the modified version of this conjecture‎. ‎in this paper‎, ‎we adopt another method to prove it‎.