A Note on Energy and Sombor Energy of Graphs

For a graph G with V (G) = {v₁, v₂, ..., v_n} and degree sequence (d_v1, d_v2, ..., d_vn), the adjacency matrix A(G) of G is a (0, 1) square matrix of order n with ij-th entry 1, if vi is adjacent to v_j and 0, otherwise. The Sombor matrix S(G) = (s_ij) is a square matrix of order n, where s_ij = ∑d²_vi + d²_vj, whenever vi is adjacent to vj, and 0, otherwise. The sum of the absolute values of the eigenvalues of A(G) is the energy, while the sum of the absolute eigenvalues of S(G) is the Sombor energy of G. In this note, we provide counter examples to the upper bound of Theorem 18 in [13] and Theorem 1 in [16].