On the First Zagreb Energy of Graphs

Researchers have been devoting themselves to the study of extended adjacency matrices, one of the many novel graph matrices that are being proposed as a potential extension of the spectral theory of classical graph matrices. The concept of the Zagreb matrix and Zagreb energy of a graph has been introduced to expand some beneficial molecular topological properties. The (first) Zagreb matrix Z(G) = (z_ij)_n×n of a graph G whose vertex v_i has degree d_i is defined by z_ij = d_i + d_j, if the vertices v_i and v_j are adjacent and z_ij = 0 otherwise. Let ζ₁, ζ₂, …, ζ_n be the Zagreb eigenvalues of Z(G) and the Zagreb energy is the sum of the absolute values of the Zagreb eigenvalues. The spectral properties of the Zagreb matrix of some classes of graphs are explored in this article. The main contribution of the article is that the Zagreb energy of a graph, obtained by means of various graph products like the strong product, corona product, Kronecker product etc. are investigated, and well-established relations are derived in terms of their base graphs.