On α-adjacency energy of graphs and Zagreb index. Issue 1 (2nd January 2021) - Pirzada, S.; Rather, Bilal A.; Ganie, Hilal A.; ul Shaban, Rezwan
Abstract: Let A ( G ) be the adjacency matrix and D ( G ) be the diagonal matrix of the vertex degrees of a simple connected graph G . Nikiforov defined the matrix A α ( G ) of the convex combinations of D ( G ) and A ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ), for 0 ≤ α ≤ 1 . If ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n are the eigenvalues of A α ( G ) (which we call α -adjacency eigenvalues of G ), the α -adjacency energy of G is defined as E A α ( G ) = ∑ i = 1 n | ρ i − 2 α m n |, where n is the order and m is the size of G . We obtain upper and lower bounds for E A α ( G ) in terms of the order n, the size m and the Zagreb index Zg ( G ) associated to the structure of G . Further, we characterize the extremal graphs attaining these bounds.