Radio labeling is an important graph labeling technique that arises from frequency assignment problems in wireless communication networks, where interference between transmitters must be minimized. The radio number of a graph represents the minimum span required to assign labels to vertices while satisfying prescribed interference constraints based on vertex distances. In this paper, we investigate the radio number of the total transformation graph of the path graph \(P_{n}^{+++}\). We study the structural properties of this transformed graph and establish lower and upper bounds for its radio number. Further, exact values are obtained for the even case, while tight bounds are derived for the odd case. The results presented in this work extend > existing research on radio labeling of transformed graphs and contribute to the study of graph labeling problems related to communication network optimization.
The binding number of a graph quantifies how strongly its vertices are tied together by their neighborhoods. In this paper, we prove that if the binding number $b > 1$, then every isolated realizing set is an isolated dominating set. We further establish several inequalities relating the binding number and the isolated domination number, and derive bounds involving structural parameters and spectral quantities. Our results highlight a close interplay between domination-type parameters and the binding number.
In a graph $G = (V, E)$, a subset $S \subseteq V$ is called a dominating set if every vertex in $V \setminus S$ is adjacent to at least one vertex in $S$. For each vertex $u \in S$, the number of edges between $u$ and vertices in $V \setminus S$ is defined as the out-degree of $u$ with respect to $S$. A dominating set $S$ is termed an out-degree equitable restrained dominating set if $S$ is a restrained dominating set, and every vertex outside $S$ is adjacent to a vertex in $S$ with the degree difference at most one. In this paper, we investigate fundamental properties, derive bounds, and present key results regarding out-degree equitable restrained domination in graphs.
A method for finding the global minimum of continuous functions is proposed. The method is based on representing the optimized parameters as approximations by trigonometric polynomials, which makes it possible to approximate functions of arbitrary complexity. The method imposes no restrictions on the objective function whatsoever - in particular, it does not require knowledge of the Lipschitz constant. The method has been tested on numerous model problems with dimensionality up to 5000 variables.
Graph theory is a fundamental field of study across various scientific disciplines, including mathematics, chemistry, security, natural sciences, and computer science. It is extensively used to solve numerous real-world problems, particularly within network theory and security frameworks. Let G = (V(G, E(G)) be a molecular graph. Then in chemical graph theory, various graph parameters, known as topological indices, have been employed to establish correlations between molecular architectures and their chemical behavior, physical properties, or biological effects. These indices are also utilized to evaluate the robustness of graph-based structures. The double hexagonal chain graphs, denoted by D-n, represent a class of polycyclic aromatic hydrocarbons. In this paper, six distinct topological indices based on reverse degrees such as the reverse Nirmala index RN(D-n), the reverse geometric-arithmetic index RGA(D-n), the reverse inverse sum index ISI(D-n), the reverse atombond connectivity index RABC(D-n), the reverse forgotten index RF(D-n) and the reverse harmonic index RH(D-n) have been computed for the double hexagonal chain graphs. Furthermore, the results have been presented via both graphical representations and detailed numerical data. Then, the relation RF(D-n) > RN(D-n) > RGA(D-n)> RH(D-n) > ISI(D-n) > RABC(D-n) is obtained as the value of n increases.
