Spectral properties of a class of unicyclic graphs

The eigenvalues of G are denoted by [Formula: see text], where n is the order of G. In particular, [Formula: see text] is called the spectral radius of G, [Formula: see text] is the least eigenvalue of G, and the spread of G is defined to be the difference between [Formula: see text] and [Formula: see text]. Let [Formula: see text] be the set of n-vertex unicyclic graphs, each of whose vertices on the unique cycle is of degree at least three. We characterize the graphs with the kth maximum spectral radius among graphs in [Formula: see text] for [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], and [Formula: see text] if [Formula: see text], and the graph with minimum least eigenvalue (maximum spread, respectively) among graphs in [Formula: see text] for [Formula: see text].