On the normalized Laplacian spectral radii of a graph and its line graph

Abstract

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.