Bounds on the spread and trace norm of the adjacency matrix of a graph

Abstract

The trace norm of the adjacency matrix A(G) of a graph G is the sum of absolute values of the eigenvalues of A(G) and it is well-known in the literature under the name energy (graph energy) of G. The spread (also known as spectral spread) of G is the difference between the largest and smallest eigenvalues of the matrix A(G). In this study, we obtain certain upper bounds for the spread of G and characterize the extremal graphs. The resulting upper bounds generalize and improve several previously known upper bounds for the spread of G [Gregory, Hershkowitz and Kirkland, The spread of the spectrum of a graph, Linear Algebra Appl., 332-334, (2001); Liu and Mu-Hou, On the spread of the spectrum of a graph, Discrete Math. 309 (2009)]. As a result of these upper bounds, we get some lower bounds for the energy of G. These lower bounds generalize and improve on certain previously established lower bounds for the enrrgy of G. Specifically, our lower bounds for the energy of G improve the lower bounds and characterize the extremal graphs obtained by [Jahanbani and Sheikholeslami, Some lower bounds on the energy of graphs in terms of spread of matrix, Mediterr. J. Math. 20 (2023)].