LAUSR

The eigenvalues of G are denoted by λ1(G),λ2(G),…,λn(G)$\lambda_{1}(G), \lambda_{2}(G), \ldots, \lambda_{n}(G)$, where n is the order of G. In particular, λ1(G)$\lambda _{1}(G)$ is called the spectral radius of G, λn(G)$\lambda_{n}(G)$ is the least eigenvalue of G, and the spread of G is defined to be the difference between λ1(G)$\lambda_{1}(G)$ and λn(G)$\lambda_{n}(G)$. Let U(n)$\mathbb{U}(n)$ 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 U(n)$\mathbb{U}(n)$ for k=1$k=1$ if n≥6$n\ge6$, k=2$k=2$ if n≥8$n\ge8$, and k=3,4,5$k=3,4,5$ if n≥10$n\ge10$, and the graph with minimum least eigenvalue (maximum spread, respectively) among graphs in U(n)$\mathbb{U}(n)$ for n≥6$n\ge6$.