Counting relations for general zagreb indices

The first and second general Zagreb indices of a graph G, with vertex set V and edge set E, are defined asMk1 = ∑v∈V d(u)k and Mk2 = ∑uv∈E[d(u) d(v)]k, where d(v) is the degree of the vertex v of G. We present combinatorial identities, relating Mk1 and Mk2 with counts of various subgraphs contained in the graph G

n-Dimensional Fractional Frequency Laplace Transform by the Inverse Difference Operator

With the study of extensive literature on the Laplace transform with one and two variables and its properties, applications are available, but there is no work on n-dimensional Laplace transform. In this research article, we define n-dimensional fractional frequency Laplace transform with shift values. Several theorems are derived with properties of the Laplace transform. The results are numerically analyzed and discussed through MATLAB

Estimating the Spread of Generalized Compartmental Model of Monkeypox Virus Using a Fuzzy Fractional Laplace Transform Method

The main objective of this work is to develop the fuzzy fractional mathematical model that will be used to examine the dynamics of monkeypox viral transmission. The proposed dynamical model consists of human and rodents individuals and this monkeypox infection model is mathematically formulated by fuzzy fractional differential equation defined in Caputo’s sense. We provide results that demonstrate the existence and uniqueness of the considered model’s solution. We observe that our results are accurate, and that our method is applicable to the fuzzy system of fractional ordinary differential equations (ODEs). Furthermore, this monkeypox virus model has been identified as a generalization of SEIQR and SEI models. The results show that keeping diseased rodents apart from the human population reduces the spread of disease. Finally, we present brief discussions and numerical simulations to illustrate our findings

Estimating the Spread of Generalized Compartmental Model of Monkeypox Virus Using a Fuzzy Fractional Laplace Transform Method

The main objective of this work is to develop the fuzzy fractional mathematical model that will be used to examine the dynamics of monkeypox viral transmission. The proposed dynamical model consists of human and rodents individuals and this monkeypox infection model is mathematically formulated by fuzzy fractional differential equation defined in Caputo’s sense. We provide results that demonstrate the existence and uniqueness of the considered model’s solution. We observe that our results are accurate, and that our method is applicable to the fuzzy system of fractional ordinary differential equations (ODEs). Furthermore, this monkeypox virus model has been identified as a generalization of SEIQR and SEI models. The results show that keeping diseased rodents apart from the human population reduces the spread of disease. Finally, we present brief discussions and numerical simulations to illustrate our findings

Higher Order Multi-Series Arising from Generalized Alpha-Difference Equation

Abstract In this paper, the authors extend the theory and m−series of the generalized difference equation to m(α)−series of its α−difference equation. We also investigate the complete and summation solutions of α-difference equation. Suitable examples are provided to illustrate the main results. Mathematics Subject Classification: 39A70, 47B39, 39A1

Symmetric Difference Operator in Quantum Calculus

The main focus of this paper is to develop certain types of fundamental theorems using q, q(α), and h difference operators. For several higher order difference equations, we get two forms of solutions: one is closed form and another is summation form. However, most authors concentrate only on the summation part. This motivates us to develop closed-form solutions, and we succeed. The key benefit of this research is finding the closed-form solutions for getting better results when compared to the summation form. The symmetric difference operator is the combination of forward and backward difference symmetric operators. Using this concept, we employ the closed and summation form for q, q(α), and h difference symmetric operators on polynomials, polynomial factorials, logarithmic functions, and products of two functions that act as a solution for symmetric difference equations. The higher order fundamental theorems of q and q(α) are difficult to find when the order becomes high. Hence, by inducing the h difference symmetric operator in q and q(α) symmetric operators, we find the solution easily and quickly. Suitable examples are given to validate our findings. In addition, we plot the figures to examine the value stability of q and q(α) difference equations