New upper bounds on Zagreb indices with given domination number

Hadi Rahbani; Hossein Abdollahzadeh Ahangar; Michael A. Henning

doi:10.1016/j.amc.2025.129815

New upper bounds on Zagreb indices with given domination number

Hadi Rahbani, Hossein Abdollahzadeh Ahangar, Michael A. Henning

Mathematics and Applied Mathematics

Babol Noshirvani University of Technology

Research output: Contribution to journal › Article › peer-review

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, deg_G(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 M₁(G)=∑v∈V(G)deg_G²(v)andM₂(G)=∑uv∈E(G)deg_G(u)deg_G(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].

Original language	English
Article number	129815
Journal	Applied Mathematics and Computation
Volume	514
DOIs	https://doi.org/10.1016/j.amc.2025.129815
Publication status	Published - 1 Apr 2026

Keywords

Domination number
First Zagreb index
Second Zagreb index
Trees

ASJC Scopus subject areas

General Computer Science
Computational Mathematics
Applied Mathematics

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10.1016/j.amc.2025.129815

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title = "New upper bounds on Zagreb indices with given domination number",

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 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 Borovi{\'c}anin and Furtula [Appl. Math. Comput. 279 (2016), 208–218].",

keywords = "Domination number, First Zagreb index, Second Zagreb index, Trees",

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year = "2026",

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doi = "10.1016/j.amc.2025.129815",

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T1 - New upper bounds on Zagreb indices with given domination number

AU - Rahbani, Hadi

AU - Abdollahzadeh Ahangar, Hossein

AU - Henning, Michael A.

PY - 2026/4/1

Y1 - 2026/4/1

N2 - 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 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].

AB - 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 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].

KW - Domination number

KW - First Zagreb index

KW - Second Zagreb index

KW - Trees

UR - https://www.scopus.com/pages/publications/105021483968

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DO - 10.1016/j.amc.2025.129815

M3 - Article

AN - SCOPUS:105021483968

SN - 0096-3003

VL - 514

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

M1 - 129815

ER -

New upper bounds on Zagreb indices with given domination number

Abstract

Keywords

ASJC Scopus subject areas

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