Some new relations between Sombor index and graph energy

Let be a graph with being vertices and being the corresponding vertex degrees. The Sombor index of is denoted by and is defined by where means there exists an edge between the vertices and . Among many other aspects of this index, the relationship of the Sombor index with adjacency graph energy has been studied by many researchers. In this paper, first, we improve the bounds of the second Zagreb index, which is better than the previous existing bounds, and using that result, we establish a relation between the Sombor index and adjacency graph energy. The extended adjacency matrix of a graph of order is the real matrix whose -th element will be if and 0 otherwise. The extended energy is the aggregate of the absolute values of the eigenvalues of this matrix. Finally, we establish a relationship between the Sombor index and the extended energy. We find an upper and a lower bound of the Sombor index in terms of the extended energy, order of the graph , size of the graph , maximum degree and minimum degree . All the relations established in this paper involve the order of the graph , size of the graph , maximum degree and minimum degree .