Arithmetic-Geometric Spectral Radius and Energy of Graphs

Let G be a graph of order n with vertex set V(G) = {v(1), v(2), ..., v(n)} and let d(i) be the degree of the vertex v(i) of G for i = 1,2, ..., n. The arithmetic-geometric adjacency matrix A(ag)(G) of G is defined so that its (i,j)-entry is equal to d(i)+d(j)/2 root d(i)+d(j), if the vertices v(i) and v(j) are adjacent, and 0 otherwise. The arithmetic-geometric spectral radius and arithmetic-geometric energy of G are the radius and energy of its arithmetic-geometric adjacency matrix, respectively. In this paper, some sharp lower and upper bounds on arithmetic-geometric radius and arithmetic-geometric energy are obtained, and the respective extremal graphs are characterized.