标题： 新的改进的图的Zagreb指标下界 显示英文标题

标题： New improved lower bounds for Zagreb indices of graphs

摘要： 本文提出了关于简单图 $G$ 的顶点的两个、三个和四个任意度数的首个一般Zagreb指数 $Z_{\alpha}(G)$的新下界。 对于特殊情况 $\alpha = 2$ 和 $\alpha = -2$，这些结果给出了首个Zagreb指数 $M_1(G)$和修改后的首个Zagreb指数 ${}^{m}M_1(G)$的更紧致的界限，从而改进了文献中的一些著名不等式。 此外，还展示了针对 $M_1(G)$ 的推导界限的一些应用，建立了第二个Zagreb指数、谱半径、Nordhaus-Gaddum型界限及其相应共指标的新界限。 显示英文摘要

摘要： This paper presents new lower bounds for the first general Zagreb index $Z_{\alpha}(G)$ involving two, three, and four arbitrary degrees of vertices of a simple graph $G$. For the special cases $\alpha = 2$ and $\alpha = -2$, the results give sharper bounds for the first Zagreb index $M_1(G)$ and the modified first Zagreb index ${}^{m}M_1(G)$, thereby improving several well-known inequalities in the literature. Furthermore, some applications of the derived bounds for $M_1(G)$ are demonstrated, establishing new bounds for the second Zagreb index, the spectral radius, Nordhaus-Gaddum type bounds, and their corresponding coindices.