The first general Zagreb index of graphs and their line graphs

Let \(\alpha \) $\alpha$ be an arbitrary real number. The first general Zagreb index \(M_\alpha (G)\) $M_{\alpha }(G)$ of a graph G is equal to the sum of the \(\alpha \) $\alpha$ th powers of the degrees of the vertices of G. Let \(\alpha \) $\alpha$ be a real number and G, a graph of order n. In this paper, we show that (1) if \(\alpha \ge 1\) $\alpha \ge 1$ and G is connected that is neither a path nor a star, then \(M_\alpha (G)\le M_\alpha (L(G))\) $M_{\alpha }(G)\le M_{\alpha }(L(G))$ ; (2) if \(0<\alpha <1\) $0<\alpha <1$ and \(\delta (G)\ge 2\) $\delta (G)\ge 2$ , then \(M_\alpha (G)\le M_\alpha (L(G))\) $M_{\alpha }(G)\le M_{\alpha }(L(G))$ with equality if and only if \(G\cong C_n\) $G\cong C_n$ ; (3) if \(\alpha \le -1\) $\alpha \le -1$ and G is a connected graph of size \(m\le n\) $m\le n$ , then \(M_\alpha (L(G))\le M_\alpha (G)\) $M_{\alpha }(L(G))\le M_{\alpha }(G)$ with equality if and only if \(G\cong C_n\) $G\cong C_n$ .