Journals Information
Mathematics and Statistics Vol. 13(1), pp. 12 - 16
DOI: 10.13189/ms.2025.130102
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Lower Bound for The Second Hyper-Zagreb Index of Trees with A Given Roman Domination Number
Waqar Ali 1, Mohamad Nazri Husin 1,*, Muhammad Faisal Nadeem 2, Muqaddas Jabin 3
1 Special Interest Group on Modeling and Data Analytics, Faculty of Computer Science and Mathematics, Universiti Malaysia Terengganu, Kuala Nerus 20130, Terengganu, Malaysia
2 Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
3 Department of Mathematics, University of Okara, N-5 Okara, 56300, Pakistan
ABSTRACT
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鈥檚 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. Future research may extend these methods to broader classes of graphs, thereby further expanding their applicability in real-world contexts.
KEYWORDS
Tree, Roman Domination Number, Second Hyper-Zagreb Index, Lower Bound
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Waqar Ali , Mohamad Nazri Husin , Muhammad Faisal Nadeem , Muqaddas Jabin , "Lower Bound for The Second Hyper-Zagreb Index of Trees with A Given Roman Domination Number," Mathematics and Statistics, Vol. 13, No. 1, pp. 12 - 16, 2025. DOI: 10.13189/ms.2025.130102.
(b). APA Format:
Waqar Ali , Mohamad Nazri Husin , Muhammad Faisal Nadeem , Muqaddas Jabin (2025). Lower Bound for The Second Hyper-Zagreb Index of Trees with A Given Roman Domination Number. Mathematics and Statistics, 13(1), 12 - 16. DOI: 10.13189/ms.2025.130102.