On the Zagreb Energy and Zagreb Estrada Index of Graphs

The (first) Zagreb matrix Z(G) = (z_ij)_n×n of a graph G whose vertex v_i has degree d_i is defined by z_ij = d_i + d_j if the vertices v_i and v_j are adjacent and z_ij = 0 otherwise. Let ζ₁, ζ₂, . . ., ζ_n be the first Zagreb eigenvalues of Z(G). The Zagreb energy ZE and the Zagreb Estrada index ZEE of a graph G are (Formula presented), respectively. Very recently, Rad et al. [N. J. Rad, A. Jahanbani, I. Gutman, Zagreb Energy and Zagreb Estrada Index of Graphs, MATCH Commun. Math. Comput. Chem. 79 (2018) 371–386] introduced and investigated the Zagreb energy and Zagreb Estrada index of a graph. We found several errors in the results of the above paper. In this paper we correct these results and some of these results are presented in a revise form. Finally, we establish some new upper and lower bounds on ZE and ZEE. Moreover, we present some novel lower and upper bounds on the spectral radius of the (first) Zagreb matrix of the graph G.