Extremal graphs with maximum complementary second Zagreb index

Hui Gao; Hui Gao

doi:10.3934/math.2025721

AIMS Mathematics

2025, Volume 10, Issue 7: 16105-16116. doi: 10.3934/math.2025721

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Extremal graphs with maximum complementary second Zagreb index

Hui Gao ^,

School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China

Received: 17 April 2025 Revised: 28 June 2025 Accepted: 03 July 2025 Published: 17 July 2025
MSC : 05C35, 05C92

The complementary second Zagreb index of a graph $ G $ is defined as $ cM_{2}(G) = \sum_{uv \in E(G)} |d_{G}(u)^{2}-d_{G}(v)^{2}| $. In this paper, we prove that a graph having maximum complementary second Zagreb index among all graphs of order $ n $ is isomorphic to $ K_{m} \vee \overline{K_{n-m}} $ for some $ 1 \leq m < \lceil \frac{n}{2} \rceil $, confirming a conjecture of Furtula and Oz.
- complementary second Zagreb index,
- extremal graph,
- mixed graph,
- orientation,
- $ cM_{2} $-maximal
Citation: Hui Gao. Extremal graphs with maximum complementary second Zagreb index[J]. AIMS Mathematics, 2025, 10(7): 16105-16116. doi: 10.3934/math.2025721

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Abstract

The complementary second Zagreb index of a graph $ G $ is defined as $ cM_{2}(G) = \sum_{uv \in E(G)} |d_{G}(u)^{2}-d_{G}(v)^{2}| $. In this paper, we prove that a graph having maximum complementary second Zagreb index among all graphs of order $ n $ is isomorphic to $ K_{m} \vee \overline{K_{n-m}} $ for some $ 1 \leq m < \lceil \frac{n}{2} \rceil $, confirming a conjecture of Furtula and Oz.

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