A Note on Energy and Sombor Energy of Graphs
Bilal Ahmad RatherDepartment of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain 15551, Abu Dhabi, UAEMuhammad Kamran SiddiquiDepartment of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain 15551, Abu Dhabi, UAE
2022en
ABI
Abstract
For a graph G with V (G) = {v1, v2, . . . , vn} and degree sequence (dv 1 , dv 2 , . . . , dv n ), 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 vj and 0, otherwise. The Sombor matrix S(G) = (sij) is a square matrix of order n, where sij = d 2 v i + d 2 v j , 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
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