On the ABC spectra radius of unicyclic graphs

The ABC matrix of a graph G is a square matrix, where the (i, j)-entry ABC(ij) = root d(i)+d(j)-2/d(i)d(j) if there is an edge between vertices i and j in G, and ABC(ij) = 0 otherwise. The ABC spectral radius of G, denoted by v(1) (G), is the largest eigenvalue of the ABC matrix of G. In this paper, we prove that for a unicyclic graph G of order n >= 4, root 2 = v(1) (C-n) <= v(1) (G) <= v(1) (S-n + e), with equality if and only if G congruent to C-n for the lower bound, and if and only if G congruent to S-n + e for the upper bound. This solves a conjecture by Ghorbani et al. (C) 2020 Elsevier Inc. All rights reserved.