Another Approach to a Conjecture about the Exponential Reduced Sombor Index of Molecular Trees

Document Type : Research Paper


Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences Ferdowsi University of Mashhad ‎ ‎P.O‎. ‎Box 1159‎, ‎Mashhad 91775‎, ‎Iran


‎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) e3 +(1/3) (n-5) e 3√2, where n≡2 (mod 3), eSored‎(T)≤(1/3) (2n+1) e3 +(1/3) (n-13) e3√2 + 3e√13 , where n≡1 (mod 3) and  eSored‎(T)≤(2/3) ne3 +(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. 


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