Maximum Zagreb indices in the class of $k$-apex trees
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The first and second Zagreb indices of a graph $G$ are defined as $M_1(G)=\sum_{v\in V} d_G(v)^2$ and $M_2(G)=\sum_{uv\in E(G)}d_G(u)\,d_G(v)\,,$ where $d_G(v)$ is the degree of the vertex $v$. $G$ is called a $k$-apex tree if $k$ is the smallest integer for which there exists a subset $X$ of $V(G)$ such that $|X|=k$ and $G-X$ is a tree. In this paper, we determine the maximum Zagreb indices in the class of all $k$-apex trees of order $n$ and characterize the corresponding extremal graphs.
Abstract
The first and second Zagreb indices of a graph $G$ are defined as $M_1(G)=\sum_{v\in V} d_G(v)^2$ and $M_2(G)=\sum_{uv\in E(G)}d_G(u)\,d_G(v)\,,$ where $d_G(v)$ is the degree of the vertex $v$. $G$ is called a $k$-apex tree if $k$ is the smallest integer for which there exists a subset $X$ of $V(G)$ such that $|X|=k$ and $G-X$ is a tree. In this paper, we determine the maximum Zagreb indices in the class of all $k$-apex trees of order $n$ and characterize the corresponding extremal graphs.
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