Iranian Journal of Mathematical Chemistry
Volume:13 Issue: 2, Spring 2022

Algebraic hyperstructures represent a natural extension of classical algebraic structures and they have many applications in various sciences. In this paper, we investigate mathematical structures of chemical reactions for arbitrary elements with four oxidation states.

‎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.

The neighborhood first Zagreb index has recently been introduced for characterizing the topological structure of molecular graphs. In the present study, we characterize the graphs having minimum neighborhood first Zagreb index in the class of unicyclic/bicyclic graphs on vertices for every fixed integer n ≥ 5.

A topological index is a numerical data which significantly correlates with the fundamental topology of a given chemical structure. The M-polynomial is a key mathematical tool to determine the degree-dependent topological indices. Very recently, the geometric-quadratic (GQ) and quadratic-geometric (QG) indices of a graph are introduced and computed their values by their respective mathematical formulas on some standard graphs and jagged-rectangle benzenoid system. In this research work, we propose M-polynomial based closed derivation formulas for determining the above two indices. In addition, we derive the GQ and QG indices for each of the abovementioned graphs by applying the derivation formulas, and also produce some fundamental relationships between the indices.

In this article, we complement the study of Pan and Li by computing the first five minimum values of the symmetric division degree (SDD) index attained by bicyclic graphs that have a perfect matching. One of our main contributions is identifying the graphs that attain the bounds. Further, we compute the upper bound of the SDD index for bicyclic graphs with a maximum degree of four, which admits a perfect matching and prove the bound is also tight by identifying the graphs that attain it.