Bounds for the largest eigenvalue of the generalized atom-bond connectivity matrix of a graph

Let [Formula: see text] be a graph with [Formula: see text] vertices, and let [Formula: see text] be the degree of its [Formula: see text]th vertex. For any real number [Formula: see text], the generalized ABC matrix of [Formula: see text] is an [Formula: see text] square matrix whose entries are [Formula: see text] if the [Formula: see text]th and the [Formula: see text]th vertex of [Formula: see text] are adjacent, and [Formula: see text] otherwise. In this paper, some lower and upper bounds for the generalized ABC-spectral radius of graphs, are obtained, in terms of minimum degree, maximum degree and the number of pendant vertices of [Formula: see text]. Graphs which attain the bounds are determined, too. Some of these bounds for the general ABC-spectral radius, when [Formula: see text], improve known results on the (usual) ABC-spectral radius.