Annals of Communications in Mathematics

Abstract. This study introduces a new probability distribution called the Cosine Exponential (CEX) Distribution, which combines the Cosine-G family of distributions with the Exponential distribution as the baseline model to create a more adaptable model. The aim is to improve modeling capabilities across various statistical applications. The paper presents expression of the density and distribution functions of the CEX model and investigates its key properties such as survival and hazard rate functions, reverse hazard function, cumulative hazard function, quantile function, moments, and moment generating function. It also outlines the methodology for estimating model parameters using maximum likelihood estimation. Through application to real datasets, the effectiveness of the proposed CEX distribution is demonstrated, showing significant enhancements over existing models. This paper highlights the potential of the CEX distribution as a robust tool for statistical modeling and analysis.

ABSTRACT. In this paper, we analyze a vector-host epidemic model with a piecewise-smooth treatment rate. The use of piecewise-smooth treatment depicts the limited medical resource situation in the community. The treatment increases linearly with infective population until the treatment capacity is reached, after which constant treatment (i.e., maximum treatment) is applied. The analysis indicates that there exists a critical value \( I_{h0}^c = \frac{b_h}{\mu_h} \) for the infective human population level \( I_{h0} \) at which the health care system reaches its capacity. We derive that when \( I_{h0} \geq I_{h0}^c \), the dynamics of the model is completely determined by the basic reproduction number \( R_0 \). When \( I_{h0} < I_{h0}^c \), the model exhibits multiple endemic equilibria.

Abstract: Let \( G \) be a graph. An Edouard Product Cordial Labeling (EPCL) of a graph \( G \) with \( |V(G)| = n \) is an injective function \( f : V(G) \rightarrow \{E_0, E_1, E_2, \ldots, E_{n-1}\} \) where \( E_i \) is the \( i \)th Edouard number \( (i = 0,1,2,3,\ldots,n) \) that induces a function \( f^* \) defined by\[f^*(uv) = (f(u)f(v)) \; (\text{mod } 2)\]for all edge \( e = uv \) such that \( |e_f^*(0) - e_f^*(1)| \leq 1 \) where \( e_f^*(0) \) is the number of vertices labeled with 0 and \( e_f^*(1) \) is the number of vertices labeled with 1. The graph that satisfies the condition of an edouard product cordial labeling is called an edouard product cordial graph (EPCG).

Abstract: Let \( W_M \) be the wheel graph on \( M \geq 4 \) vertices and let \( \overline{K_n} \) be the independent graph on \( n \geq 1 \) vertices. We study the corona product \( W_M \circ K_n \) and obtain an explicit formula for its pendant domination polynomial. The computation starts from the domination polynomial and subtracts a correction term that counts dominating sets whose induced subgraph contains no vertex of degree 1. For the wheel, the correction term reduces to counting subsets of the rim cycle for which the selected rim vertices are not isolated on the rim. We also determine the pendant domination number for this family.

Abstract: Hop domination was introduced as a distance-two analogue of domination and has been studied extensively in recent years. A secure hop dominating set, recently introduced, models a single adversarial attack at an unoccupied vertex (a vertex not in the current guard set) that can be defended by relocating one guard at distance two while preserving hop domination. Motivated by finite-order (multi-step) protection in classical secure domination, we introduce \( t \)-secure hop dominating sets (\( t \in \mathbb{N}_0 \)), in which an adversary may launch a sequence of at most \( t \) attacks, each at a currently unoccupied vertex, and the defender responds by sequentially relocating one guard at distance two after each attack while maintaining hop domination throughout. Our main contribution is an exact correspondence: \( t \)-secure hop domination in a graph \( G \) is equivalent to smart \( t \)-secure domination in the hop graph \( H(G) \). This yields structural properties (monotonicity and additivity over components) and exact values for several graph families, including complete multipartite graphs, stars, paths, and cycles. In particular, we obtain closed formulas for \( \gamma_{sh,t}(P_n) \) and \( \gamma_{sh,t}(C_n) \) for all \( t \in \mathbb{N}_0 \), with explicit small-\( n \) exceptions in the cycle case.

Abstract:An injective function \( f : V(G) \rightarrow \{L_1, L_2, \ldots, L_n\} \), where \( L_i \) is the \( i \)th Lucas number, is called a Lucas product cordial labeling if the induced function satisfies \( |e_f^*(0) - e_f^*(1)| \leq 1 \). A graph which admits Lucas product cordial labeling is called Lucas product cordial graph. In this paper, we determined the Lucas Product Cordial Labeling of Quadrilateral Snake Graph Qn, Cycle Quadrilateral Snake Graph CQn, and Alternate Triangular Snake Graph A(Tn).

Abstract: In this paper we introduce a new orbit-based contractive framework in the setting of \( G \)-metric spaces, called \( (m,\alpha) \) \( G \)-path-averaged (\( G \)-PA) contractions with \( m \geq 2 \). This notion extends Fabião’s path-averaged contractions to the triadic geometry of Mustafa–Sims \( G \)-metrics and is designed to avoid collapse to pointwise contractivity. For a \( G \)-continuous self-map on a complete \( G \)-metric space, we establish existence and uniqueness of a fixed point and prove that the Picard iteration converges to it in the sense of \( G \)-convergence. Moreover, we derive explicit quantitative estimates, including a posteriori and a priori geometric error bounds for the iterates. We also relate the new class to the induced metric \( d_G \), showing that every \( G \)-PA contraction yields a path-averaged contraction on \( (X, d_G) \), and we provide examples demonstrating that the \( G \)-PA class can be strictly larger than the Banach-type contraction class. Finally, we obtain multi-step (\( t \)-point) fixed point and convergence results by embedding the recursion into a shift map on the product space \( (X^t, \sigma^t) \) and applying the single-valued theory.

Abstract: Let \( p \) be an odd prime and let \( F_i \) be the \( i \)th \( (a,b) \)-Fibonacci number with initial values \( F_0 = a \) and \( F_1 = b \). For a simple connected graph \( G = (V,E) \), define a bijective function \( f : V(G) \rightarrow \{0,1,\ldots,|V|-1\} \). If the induced function \( f_p^* : E(G) \rightarrow \{0,1\} \), defined by \( f_p^*(uv) = \frac{1+((F_{f(u)}+F_{f(v)})/p)}{2} \) whenever \( F_{f(u)} + F_{f(v)} \not\equiv 0 \; (\text{mod } p) \) and \( f_p^*(uv) = 0 \) whenever \( F_{f(u)} + F_{f(v)} \equiv 0 \; (\text{mod } p) \), satisfies the condition \( |e_f^*(0) - e_f^*(1)| \leq 1 \) where \( e_f^*(i) \) is the number of edges labeled \( i \) (\( i = 0,1 \)), then \( f \) is called a \( (a,b) \)-Fibonacci-Legendre cordial labeling modulo \( p \). In this paper, the \( (a,b) \)-Fibonacci-Legendre cordial labeling of path graphs, star graphs, wheel graphs, and graphs under the operations join, corona, lexicographic product, cartesian product, tensor product, and strong product is explored in relation to \( k \)-Pisano-Legendre primes relative to \( (a,b) \). We also present some properties of \( k \)-Pisano-Legendre primes relative to \( (a,b) \) and numerical observations on its distribution, leading to several conjectures concerning their density and growth behavior.

Abstract: In the paper, we investigate the Hyers–Ulam stability theorem of an Euler–Lagrange additive functional inequality\[\left\| \sum_{j=1}^{n} \left( \sum_{k=j}^{n} \lambda_k \right) f(x_j - x_{j-1}) \right\| \leq \left\| f\left( \sum_{j=1}^{n} \lambda_j x_j \right) \right\| + \varphi(x_1, \ldots, x_n),\]where \( x_0 = 0 \), \( n \geq 3 \), subject to control function \( \varphi \) in (non-Archimedean) Banach spaces.