Iranian Journal of Mathematical Chemistry
Volume:14 Issue: 4, Autumn 2023

‎Let $G=(V,E)$ be a graph‎. ‎The entire Sombor index of graph ‎$‎G‎$‎, $ SO^\varepsilon(G) $ is defined as the sum of the terms‎‎$\sqrt{d_{G}^2(a)+d_{G}^2(b)}$‎, ‎where $a$ is either adjacent to or incident with $b$ and‎‎$a,b\in V\cup E$‎.‎It is known that if $T$ is a tree of order $n$‎, ‎then $SO^\varepsilon(T)\ge 6\sqrt{5}+8(n-3)\sqrt{2}$‎. ‎We improve this result and establish best lower bounds on the entire Sombor index with given vertices number and maximum degree‎. ‎Also‎, ‎we determine the extremal trees achieve these bounds.

‎The main goal of this research work is to provide a numerical technique based on choosing a set of basis functions for handling the third-order time-fractional Korteweg–De Vries Burgers' equation‎. ‎The trial functions are selected for the shifted second-kind Chebyshev polynomials (S2KCPs) compatible with the problem's governing initial and boundary conditions‎. ‎The spectral tau method transforms the equation and its underlying conditions into a nonlinear system of algebraic equations that can be efficiently numerically inverted with the standard Newton's iterative procedures after the approximate solutions have been expressed as a double expansion of the two chosen basis functions‎. ‎The truncation error is estimated‎. ‎Various numerical examples are displayed together with comparisons to other approaches in the literature to show the applicability and accuracy of the provided methodology‎. ‎Different numerical models are displayed and compared to other methods in the literature‎.

‎A self-loop graph $G_S$ is a simple graph $G$ obtained by attaching loops at $S \subseteq V(G).$ To such $G_S$ an Euclidean metric function is assigned to its vertices‎, ‎forming the so-called Sombor matrix‎. ‎In this paper‎, ‎we derive two summation formulas for the spectrum of the Sombor matrix associated with $G_S,$ for which a Forgotten-like index arises‎. ‎We explicitly study the Sombor energy $\cE_{SO}$ of complete graphs with self-loops $(K_n)_S,$ as the sum of the absolute value of the difference of its Sombor eigenvalues and an averaged trace‎. ‎The behavior of this energy and its change for a large number of vertices $n$ and loops $\sigma$ is then studied‎. ‎Surprisingly‎, ‎the constant $4\sqrt{2}$ is obtained repeatedly in several scenarios‎, ‎yielding a quantization of the energy change of 1 loop for large $n$ and $\sigma$‎.‎Finally‎, ‎we provide a McClelland-type and determinantal-type upper and lower bounds for $\cE_{SO}(G_S),$ which generalizes several bounds in the literature‎.

‎The space fractional PDEs (SFPDEs) have attracted a lot of attention‎. ‎Developing high-order and stable numerical algorithms for them is the main aim of most researchers‎. ‎This research work presents a fractional spectral collocation method to solve the fractional models with space fractional derivative which is defined based upon the Riesz derivative‎. ‎First‎, ‎a second-order difference formulation is used to approximate the time derivative‎. ‎The stability property and convergence order of the semi-discrete scheme are analyzed‎. ‎Then‎, ‎the fractional spectral collocation method based on the fractional Jacobi polynomials is employed to discrete the spatial variable‎. ‎In the numerical results‎, ‎the effect of fractional order is studied‎.

‎Topological indices are numerical parameters for understanding the fundamental topology of chemical structures that correlate with the quantitative structure-property relationship (QSPR)‎ / ‎quantitative structure-activity relationship (QSAR) of chemical compounds‎. ‎The M-polynomial is a modern mathematical approach to finding the degree-based topological indices of molecular graphs‎.‎Several graph assets have been employed to discriminate the construction of entropy measures from the molecular graph of a chemical compound‎. ‎Graph entropies have evolved as information-theoretic tools to investigate the structural information of a molecular graph‎. ‎The possible applications of graph entropy measures in chemistry‎, ‎biology and discrete mathematics have drawn the attention of researchers‎. ‎In this research work‎, ‎we compute the Nirmala index‎, ‎first and second inverse Nirmala index for silicon carbide network $Si_{2}C_{3}\textit{-I}[p,q]$ with the help of its M-polynomial‎. ‎Further‎, ‎we introduce the concept of Nirmala indices-based entropy measure and enumerate them for the above-said network‎. ‎Additionally‎, ‎the comparison and correlation between the Nirmala indices and their associated entropy measures are presented through numerical computation and graphical approaches‎. ‎Following that‎, ‎curve fitting and correlation analysis are performed to investigate the relationship between the Nirmala indices and corresponding entropy measures.