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New upper bounds on Zagreb indices with given domination number

Author

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  • Rahbani, Hadi
  • Abdollahzadeh Ahangar, Hossein
  • Henning, Michael A.

Abstract

A set D of vertices in a graph G is a dominating set of G if every vertex not in D is adjacent to a vertex in D. The domination number, γ(G), is the minimum cardinality of a dominating set of G. The degree, degG(v), of a vertex v in G is the number of vertices adjacent to v in G. The first Zagreb index, M1(G), and the second Zagreb index, M2(G)), of G are defined by(1)M1(G)=∑v∈V(G)degG2(v)andM2(G)=∑uv∈E(G)degG(u)degG(v),respectively. We obtain new upper bounds for the first and second Zagreb indices of a tree in terms of the its order, the number of leaves and the domination number, and we characterize the extremal trees that achieve equality in the obtained bounds. These results improve results of Borovićanin and Furtula [Appl. Math. Comput. 279 (2016), 208–218].

Suggested Citation

  • Rahbani, Hadi & Abdollahzadeh Ahangar, Hossein & Henning, Michael A., 2026. "New upper bounds on Zagreb indices with given domination number," Applied Mathematics and Computation, Elsevier, vol. 514(C).
    • Handle: RePEc:eee:apmaco:v:514:y:2026:i:c:s0096300325005405
    DOI: 10.1016/j.amc.2025.129815
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    References listed on IDEAS

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    1. Wyatt J. Desormeaux & Teresa W. Haynes & Michael A. Henning, 2013. "Edge lifting and total domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 25(1), pages 47-59, January.
    2. Borovićanin, Bojana & Furtula, Boris, 2016. "On extremal Zagreb indices of trees with given domination number," Applied Mathematics and Computation, Elsevier, vol. 279(C), pages 208-218.
    3. Raza, Zahid & Akhter, Shehnaz, 2023. "On maximum Zagreb connection indices for trees with fixed domination number," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
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