Ordering trees by their ABC spectral radii

Let G = (V, E) be a connected graph, where V = {v1, v2, …, vn}. Let di denote the degree of vertex vi. The ABC matrix of G is defined as M(G) = (mij)n × n, where mij=di+dj−2/didj if vivj ∈ E, and 0 otherwise. The ABC spectral radius of G is the largest eigenvalue of M(G). In the present paper, two graph perturbations with respect to ABC spectral radius are established. By applying these perturbations, the trees with the third, fourth, and fifth largest ABC spectral radii are determined. Two graph perturbations with respect to ABC spectral radius are established. Consequently, the trees of order at least 10 with the 1st to 5th largest ABC spectral radii are determined. We wonder how the “Ruler Theorem” holds for ABC spectral radius of trees.