On the Zagreb matrix and Zagreb energy

Let G be a graph of order n, and du be the degree of vertex u in a graph G. The Zagreb matrix of G is the square matrix of order n whose (i,j)-entry is equal to d(u) + d(v) if the vertex u and the vertex v of G are adjacent, and 0 otherwise. This matrix, related closely to the Zagreb (abbreviated as Z) index. The Zagreb eigenvalues of G are the eigenvalues of its Z matrix and the Zagreb energy of G is the sum of the absolute values of its Zagreb eigenvalues. In this paper, we obtain some new results for the eigenvalues of the Zagreb matrix, the spread of the Zagreb matrix. Moreover, we obtain some new bounds for the Zagreb energy of a graph.