Extremal First Leap Zagreb Index of $k$-Generalized Quasi-Trees
Received:March 26, 2021  Revised:September 22, 2021
Key Word: $k$-generalized quasi-trees   the first leap Zagreb index   $2$-distance degree
Fund ProjectL:Supported by the Foundation of Henan Department of Science and Technology (Grant No.182102310830), the Foundation of Henan University of Engineering (Grant No.D2016018) and the Foundation of Henan Educational Committee (Grant Nos.20A110016; 2020GGJS239).
 Author Name Affiliation Pei SUN School of Mathematics, Zhengzhou University of Aeronautics, Henan 450046, P. R. China Kai LIU College of Sciences, Henan University of Engineering, Henan 451191, P. R. China
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For a graph $G$, the first leap Zagreb index is defined as $LM_1(G)=\sum_{v\in V(G)}d_2(v/G)^2$, where $d_2(v/G)$ is the $2$-distance degree of a vertex $v$ in $G$. Let $\mathcal{QT}^{(k)}(n)$ be the set of $k$-generalized quasi-trees with $n$ vertices. In this paper, we determine the extremal elements from the set $\mathcal{QT}^{(k)}(n)$ with respect to the first leap Zagreb index.