Lower Bound for The Second Hyper-Zagreb Index of Trees with A Given Roman Domination Number

作者：Waqar Ali, Mohamad Nazri Husin, Muhammad Faisal Nadeem, Muqaddas Jabin · 发表于：Mathematics and Statistics · 年份：2025 · DOI：10.13189/ms.2025.130102 · 被引用次数：4 · 研究领域：Graph theory and applications、Advanced Graph Theory Research、Matrix Theory and Algorithms

Graph theory plays a crucial role in understanding the structural properties of molecular and network systems. One of the significant topological indices used in this domain is the second Hyper-Zagreb index ( ), which is computed by summing the degrees of adjacent vertices and in a molecular graph and squaring the result. This index provides valuable insights into the graph’s complexity and has chemistry, physics, and network analysis applications. Another important concept in graph theory is the Roman dominating number (RDN), defined as a function : , where is the set of vertices. The RDN must satisfy the condition that for every vertex with , there exists an adjacent vertex with , ensuring that all vertices are strategically covered. The RDN, denoted by , is the minimum total weight assigned by the RDN across all vertices and is critical for optimizing network security, resource allocation, and fault tolerance in various systems. This paper aims to bridge the gap between these two areas by establishing a lower bound on the characterized by vertices and their corresponding . Our findings reveal new insights into the interplay between these graph parameters, offering enhanced tools for precise analysis in molecular chemistry and theoretical network sciences. The derived bounds have significant implications for improving the design and resilience of complex systems, particularly in scenarios where efficient resource deployment and stability are paramount. F...