For a simple graph G with vertex set \( \{v_{1},\dots ,v_{n}\} \) and degree sequence \( \{d_{1},\dots ,d_{n}\} \) , the Sombor matrix S(G) of G is an \( n\times n \) matrix, whose (i, j) -th entry is \( \sqrt{d_{i}^{2}+d_{j}^{2}} \) , if \(v_{i}\) and \(v_{j}\) are adjacent and 0, otherwise. The multi-set of the eigenvalues of S(G) is known as the Sombor spectrum of G, denoted by \( \mu _{1}\ge \mu _{2}\ge \dots \ge \mu _{n} \) , where \( \mu _{1} \) is the Sombor spectral radius of G. The absolute sum of the Sombor eigenvalues is known as the trace norm (Sombor energy) of G. The spectral spread of S(G) is defined by \(s(S(G))=\mu _{1}-\mu _{n}\) . In this article, we establish various sharp bounds for s(S(G)) in terms of various graph parameters like order, Schatten norm, Frobenius norm, trace norm, Forgotten topological index, Sombor index and many other invariants. We give complete characterization of the extremal graphs attaining these bounds. As a consequence of s(S(G)) , we present the bounds on the trace norm of S(G) along with the graphs attaining them.
