The first general Zagreb index \(M^{\alpha }_{1}(G)\) of a graph G is equal to the sum of the \(\alpha \)th powers of the vertex degrees of G. For \(\alpha \ge 0\) and \(k \ge 1\), we obtain the lower and upper bounds for \(M^{\alpha }_{1}(G)\) and \(M^{\alpha }_{1}(L(G))\) in terms of order, size, minimum/maximum vertex degrees and minimal non-pendant vertex degree using some classical inequalities and majorization technique, where L(G) is the line graph of G. Also, we obtain some bounds and exact values of \(M^{\alpha }_{1}(J(G))\) and \(M^{\alpha }_{1}(L^{k}(G))\), where J(G) is a jump graph (complement of a line graph) and \(L^{k}(G)\) is an iterated line graph of a graph G.

在第一普通萨格勒布指数\（M ^ {\阿尔法} _ {1}（G）\）的曲线图的ģ等于总和\（\阿尔法\）个顶点度的权力ģ。对于\（\ alpha \ ge 0 \）和\（k \ ge 1 \），我们获得\（M ^ {\ alpha __1 {G} \）和\（M ^ {\ alpha} _ {1}（L（G））\）在顺序，大小，最小/最大顶点度和最小非垂线顶点度方面使用一些经典的不等式和主化技术，其中L（G）是G的折线图。同样，我们获得\（M ^ {\ alpha} _ {1}（J（G））\）的一些边界和精确值和\（M ^ {\ alpha} _ {1}（L ^ {k}（G））\），其中J（G）是跳转图（折线图的补全），而\（L ^ {k} （G）\）是图G的迭代线图。