In this paper, we establish extremal results and bounds of the atom-bond connectivity index with the arithmetic-geometric index given the minimum and maximum degrees of a graph. We have shown that regular graphs correspond to extremal graphs.

This paper is concerned with a model of disease transmission in predator-prey, predator cannibalism, and harvesting. We demonstrate that the model's solution is bounded and positive. Then, we investigate each potential equilibrium point's existence and stability. The local stability of the model around each equilibrium point is studied by the linearizing the system using Jacobian Matrix, while the global stability is performed by defining a Lyapunov function. The model has six equilibria, which are conditionally locally asymptotically stable. Global stability analysis performed shows that all equilibria are conditionally globally asymptotically stable. To support our analytical findings, we also perform numerical simulations.

In this paper, we study the Diophantine equation a^x+(a+5b)^y=z^2 when a\equiv 1\left(\bmod 5\right) and b is a positive integer. We establish that the equation has no solutions in positive integers x,y and z. We start with the Diophantine equation p^x+(p+5a)^y=z^2 where p and p+5a are both primes and p\equiv 1\left(\bmod 5\right) and a is a positive integer.

The chromatic polynomial P(G,t) related to graphs is a major area of algebraic graph theory. These polynomials allow graphs to be extracted using algebraic techniques and contain important information about their structure and properties. In particular, the coefficients, roots, and derivatives of graphing polynomials at certain points and their values are frequently interpreted in important ways. A graph with vertices V and a chromatic polynomial chi_{G}(n) is called G. In the paper, we discover some properties by defining h_{G}(z) as the generating function of the chromatic polynomial of the graph along with properties of the roots. A real number t is called a chromatic root of the generating function for the chromatic polynomial of a graph G if h(G, t) = 0. Since the chromatic polynomial enumerates the appropriate colors of a graph, it is natural to see how color roots behave in light of the assumptions related to them.

In this paper, we consider a certain inequality for positively definite Hermitian matrices. We prove this inequality using auxiliary inequalities such as Minkowski's inequality and the AM-GM inequality.

The Multivariate Spatial Durbin Model (MSDM) is a significant advance in spatial econometrics, very relevant in the context of research problems. This model extends spatial analysis by capturing the complexity and dynamism of interactions between variables in a spatial context that is often ignored by classical spatial models. Furthermore, this article aims to estimate the parameters of MSDM model applied to large and complex data sets through Monte Carlo simulations. This model was then estimated using Maximum Likelihood Estimation (MLE), and to test the accuracy of the model using the Maximum Likelihood Ratio Test (MLRT) with a computational approach. The research results show that the MSDM model para-meter estimates are accurate as indicated by an accuracy value that is smaller than the 5% significance level. The model becomes more efficient as the sample size increases.

In this paper, we establish some properties of convex accessibility polynomial of graphs. Moreover, we provide results on the convex accessibility polynomial of star graphs and complete bipartite graphs.