Normalized Laplacian eigenvalues are very popular in spectral graph theory. The normalized Laplacian spectral radius \(\rho _1(G)\) of a graph G is the largest eigenvalue of normalized Laplacian matrix of G. In this paper, we determine the extremal graph for the minimum normalized Laplacian spectral radii of nearly complete graphs. We present several lower bounds on \(\rho _1(G)\) in terms of graph parameters and characterize the extremal graphs. Still, there is no result on the normalized Laplacian eigenvalues of line graphs. Here, we obtain sharp lower bounds on the normalized Laplacian spectral radii of line graphs. Moreover, we compare \(\rho _1(G)\) and \(\rho _1(L_G)\)\((L_G \text{ is } \text{ the } \text{ line } \text{ graph } \text{ of } \)G) in some class of graphs as they are incomparable in the general case. Finally, we present a relation on the normalized Laplacian spectral radii of a graph and its line graph.

归一化的拉普拉斯特征值在频谱图理论中非常流行。归一化的拉普拉斯谱半径\（\ RHO _1（G）\）的曲线图的ģ是归一化的拉普拉斯矩阵的最大特征值G ^。在本文中，我们确定了几乎完整图的最小归一化拉普拉斯谱半径的极值图。我们根据图形参数给出\（\ rho _1（G）\）的几个下界，并刻画极值图。折线图的归一化Laplacian特征值仍然没有结果。在这里，我们在折线图的归一化拉普拉斯光谱半径上获得了尖锐的下界。此外，我们比较\（\ rho _1（G）\）和\（\ rho _1（L_G）\）\（（L_G \ text {是} \ text {the} \ text {line} \ text {graph} \ text {of} \） G）在某些类图中，因为它们通常是无法比拟的。最后，我们给出了图及其线图的归一化拉普拉斯谱半径的关系。