On the normalized Laplacian energy and general Randić index R-1 of graphs

In this paper, we consider the energy of a simple graph with respect to its normalized Laplacian eigenvalues, which we call the L-energy. Over graphs of order n that contain no isolated vertices, we characterize the graphs with minimal L-energy of 2 and maximal L-energy of 2⌊n/2⌋. We provide upper and lower bounds for L-energy based on its general Randić index R-1(G). We highlight known results for R-1(G), most of which assume G is a tree. We extend an upper bound of R-1(G) known for trees to connected graphs. We provide bounds on the L-energy in terms of other parameters, one of which is the energy with respect to the adjacency matrix. Finally, we discuss the maximum change of L-energy and R-1(G) upon edge deletion.