A finite, connected simple graph \(G\) is a geodetic graph if and only if for each pair of vertices \(v_i, v_j\) there exists a unique distance path (or unique shortest \(v_iv_j\)-path). The insertion of vertices in an edge or edges of a non-geodetic graph \(G\) to, if possible, obtain a resultant geodetic graph is called geodetication of the graph \(G\). The paper introduces two new graph parameters generally called the Ruv\(\acute{e}\) numbers of a graph. The Ruv\(\acute{e}\) numbers of \(G\) are denoted by \(\rho_1(G)\) and \(\rho_2(G)\) respectively, and \(\rho_1(G) = \rho_2(G) = 0\) if and only if \(G\) is geodetic. Furthermore, for some graphs the parameter, \(\rho_1(G) \to \infty\). The latter graphs \(G\) do not permit geodetication in respect of \(\rho_1(G)\). It is evident that geodetication presents various challenging minimization problems. The core field of application will be, restricting graphs to distance path uniqueness. Intuitive applications are foreseen in military science, IT anti-hacking coding and predictive flow through networks.

The anchored Dyck words of length \(n=2k+1\) (obtained by prefixing a 0-bit to each Dyck word of length \(2k\) and used to reinterpret the Hamilton cycles in the odd graph \(O_k\) and the middle-levels graph \(M_k\) found by M\”utze et al.) represent in \(O_k\) (resp., \(M_k\)) the cycles of an \(n\)- (resp., \(2n\)-) 2-factor and its cyclic (resp., dihedral) vertex classes, and are equivalent to Dyck-nest signatures. A sequence is obtained by updating these signatures according to the depth-first order of a tree of restricted growth strings (RGS’s), reducing the RGS-generation of Dyck words by collapsing to a single update the time-consuming \(i\)-nested castling used to reach each non-root Dyck word or Dyck nest. This update is universal, for it does not depend on \(k\).

In this paper, we introduce the concept of Sequential Henstock Stieltjes integral for interval valued functions and prove some properties of this integral.

In this note, we show how a combinatorial identity of Frisch can be applied to prove and generalize some well-known identities involving harmonic numbers. We also present some combinatorial identities involving odd harmonic numbers which can be inferred straightforwardly from our results.

In the present investigation, the authors introduce a new class of multivalent analytic functions defined by an extended Salagean differential operator. Coefficient estimates, growth and distortion theorems for this class of functions are established. For this class, we also drive radius of starlikeness. Furthermore, the integral transforms of the class are obtained.

This study looks at the worldwide behavior of a monkeypox epidemic model that includes the impact of vaccination. A mathematical model is created to analyse the vaccine impact, assuming that immunisation is administered to the susceptible population. The system’s dynamics are determined by the fundamental reproduction number, R₀. When R₀ < 1, the illness is expected to be eradicated, as evidenced by the disease-free equilibrium’s global asymptotic stability. When R₀ > 1, the illness continues and creates a globally stable endemic equilibrium. Furthermore, we investigate the existence of traveling wave solutions, demonstrating that (i) a minimal wave speed, designated as c^* > 0, exists when R₀ > 1; (ii) when R₀ ≤ 1, no nontrivial traveling wave solution exists. Additionally, for wave speeds c < c^*, no nontrivial traveling wave solution is found, whereas when c ≥ c^*, the system admits a nontrivial traveling wave solution with speed c. Numerical simulations are performed to further validate these theoretical results, confirming both the stability of the equilibrium points and the traveling wave solutions.

This study employs dynamic modeling and simulation to provide theoretical insights into the systemic behaviors underlying diphtheria pathogenesis. A Boolean network model was developed to formalize the hypothesized interactions among eight genes identified in the literature as central to toxin production, immune response, and disease transmission. Computational exploration of the state-space dynamics within this model revealed three distinct attractors, each hypothesized to represent key disease states. Structural analysis of these attractors and their basins of attraction offered insights into network architectures potentially responsible for bistable switches between chronic infection and recovery, endogenous inflammatory oscillations reflective of periodic fever cycles, and modular topologies enabling alternative developmental pathways. These findings demonstrate the utility of Boolean modeling in uncovering organizing principles—such as periodicity, bistability, and evolvability—that govern disease emergence in complex systems. The study highlights testable network signatures that could refine our understanding of diphtheria and similar pathologies, and while preliminary, it underscores the potential of iterative computational and experimental approaches to inform more effective control strategies.

This paper derives some new Hermite-Hadamard inequality and its different product versions, along with interesting non-trivial examples and remarks. Furthermore, we apply some of our results to special means as an application.

The aim of this work is to present an efficient modification of the Adomian Decomposition Method (ADM) for solving third-order ordinary differential equations with constant coefficients. The proposed approach is applicable to both linear and nonlinear problems. To demonstrate the effectiveness of the method, several examples are provided, showcasing its capability to handle both linear and nonlinear ordinary differential equations.