On Eigenvalues and Energy of Geometric–Arithmetic Matrix of Graphs

Let G be a graph with vertex set { v₁, v₂, … , v_n} and let d_i denote the degree of vertex v_i. The geometric–arithmetic matrix GA(G) of G is indexed by the vertices of G, whose (i, j) -th entry is 2didjdi+dj if the vertices i and j are adjacent and 0 otherwise. The multi-set of eigenvalues of GA(G) is known as the geometric–arithmetic spectrum of G. In this article, we obtain geometric–arithmetic spectra of various families of graphs and characterize the connected graphs with two and three distinct GA eigenvalues. We obtain several upper and lower bounds for the geometric arithmetic energy and characterize the graphs attaining such bounds.