Annals of Communications in Mathematics

Abstract: A total product cordial labeling of a graph \( G \) is a function \( f : V \rightarrow \{0,1\} \). For each \( xy \), assign the label \( f(x)f(y) \); \( f \) is called total product cordial labeling of \( G \) if it satisfies the condition that \( |v_f(0)+e_f(0)-v_f(1)-e_f(1)| \leq 1 \) where \( v_f(i) \) and \( e_f(i) \) denote the set of vertices and edges which are labeled with \( i = 0,1 \), respectively. A graph with a total product cordial labeling defined on it is called a total product cordial graph. In this paper, we determined the total product cordial labeling of the snake graphs \( T_n \), \( A(T_n) \), \( D(T_n) \), \( DA(T_n) \), \( Q_n \), \( A(Q_n) \), \( D(Q_n) \), and \( DA(Q_n) \).

Abstract The Hilbert integral inequality is a well-known and widely studied result in analysis that has inspired many refinements and modifications. In this paper, we present a new logarithmic modification of this inequality. Our approach is based on a trigonometric method that offers a fresh perspective on existing standard techniques. As a consequence, we also derive another integral inequality. All arguments are presented in full detail.

Abstract The Hilbert integral inequality is a well-known and widely studied result in analysis that has inspired many refinements and modifications. In this paper, we present a new logarithmic modification of this inequality. Our approach is based on a trigonometric method that offers a fresh perspective on existing standard techniques. As a consequence, we also derive another integral inequality. All arguments are presented in full detail.

Abstract The distinguishing number is an important invariant used to measure the extent to which symmetries of graphs and permutation group actions can be broken by vertex labelings. In this paper, we investigate distinguishing labelings arising from permutation group actions with particular emphasis on Cartesian power constructions and wreath product actions. We establish structural bounds for distinguishing numbers in terms of orbit structure, stabilizers, and base size of permutation groups. Furthermore, we analyze the behavior of distinguishing numbers under Cartesian powers of sets and derive bounds for wreath product actions of the form \( G \wr S_m \) acting on \( X^m \). Several examples involving symmetric groups are presented to illustrate the theoretical results. These results contribute to a deeper understanding of symmetry breaking in permutation group actions and provide new insights into the interplay between distinguishing numbers and algebraic structures arising from wreath products.

ABSTRACT. Let \( G = (V(G), E(G)) \) be a simple non-complete graph and let \( \xi : V \rightarrow \{0,1,2\} \) be a hop Roman dominating function (HRDF) on \( G \). For each \( j \in \{0,1,2\} \), let \( V_j = \{x \in V(G) : \xi(x) = j\} \). Then \( \xi = (V_0, V_1, V_2) \). A function \( \xi \) is an interior hop Roman dominating function (InHRDF) on \( G \) if for each \( v \in V_0 \), there exists \( u \in V_2 \) such that \( d_G(u,v) = 2 \), and either \( V_1 = V(G) \) or for every \( v \in V_2 \), \( v \) is an interior vertex of \( G \). The weight of InHRDF \( \xi \) is denoted by \( \omega_G^{\text{InhR}}(\xi) \) and is defined as \( \omega_G^{\text{InhR}}(\xi) = \sum_{u \in V(G)} \xi(u) = |V_1| + 2|V_2| \). The minimum weight of an InHRDF \( \xi \) on \( G \), denoted as \( \gamma_{\text{InhR}}(G) = \min \{ \omega_G^{\text{InhR}}(\xi) : \xi \text{ is an InHRDF on } G \} \), is called the interior hop Roman domination number. Every InHRDF \( \xi \) on \( G \) satisfying the condition \( \omega_G^{\text{InhR}}(\xi) = \gamma_{\text{InhR}}(G) \) is called a \( \gamma_{\text{InhR}} \)-function on \( G \). In this paper, we investigate a new restricted parameter of a hop Roman dominating function in graphs called the interior hop Roman domination and present some combinatorial results.

ABSTRACT. In this study, we investigate specific kernel structures within interval valued topological spaces. We introduce the notions of interval-valued Λ-sets and interval-valued λ-closed sets and discuss their essential properties. The connections between these concepts and existing notions in interval-valued topology are examined in detail. Various characterizations and foundational results are presented to clarify their structural behavior. This work aims to enhance and extend the theoretical framework of interval-valued topology.

ABSTRACT. This paper introduces statistical gauge convergence as a refinement of statistical convergence in metric spaces, where deviations from the limit are controlled by positive continuous functions rather than fixed constants. We provide equivalent density based characterizations and examine their relationship with both classical and statistical convergence, showing that the corresponding implications are strict in general. Further more, we investigate the topology generated by this convergence and prove that it is typically finer than the underlying metric topology. Several examples are included to clarify the hierarchical structure among the considered notions of convergence.

ABSTRACT. Let \( G = (V(G), E(G)) \) be a connected graph and let \( f : V(G) \rightarrow \{0,1,2\} \) be a hop Roman dominating function (HRDF) on \( G \). If for each \( k \in \{0,1,2\} \), \( V_k = \{x \in V(G) : f(x) = k\} \), then \( f = (V_0, V_1, V_2) \). A function \( f \) is an outer-convex hop Roman dominating function (OConHRDF) on \( G \) provided that for every \( v \in V_0 \), there exists \( u \in V_2 \) such that \( v \in N_G^2(u) \) and \( V_0 \) is a convex set. The weight of OConHRDF \( f \) on \( G \) is denoted by \( \widetilde{\omega}_G^{conhR}(f) \) and is defined as \( \widetilde{\omega}_G^{conhR}(f) = \sum_{v \in V(G)} f(v) \). The smallest weight of an OConHRDF \( f \) on \( G \), denoted by \( \widetilde{\gamma}_{conhR}(G) \), is called the outer-convex hop Roman domination number, which can be written as \( \widetilde{\gamma}_{conhR}(G) = \min \{ \widetilde{\omega}_G^{conhR}(f) : f \text{ is an OConHRDF on } G \} \). Every OConHRDF \( f \) on \( G \) satisfying the condition \( \widetilde{\omega}_G^{conhR}(f) = \widetilde{\gamma}_{conhR}(G) \) is so-called a \( \widetilde{\gamma}_{conhR} \)-function on \( G \). This paper introduces a new parameter of a hop Roman dominating function in graphs, called outer-convex hop Roman dominating function and presents initial investigation.