Spectral properties of a class of unicyclic graphs - Zhibin Du (Journal of Inequalities and Applications, 2017)
Abstract The eigenvalues of G are denoted by λ 1 ( G ) , λ 2 ( G ) , … , λ n ( G ) $λ_{1}(G), λ_{2}(G), …, λ_{n}(G)$ , where n is the order of G. In particular, λ 1 ( G ) $λ _{1}(G)$ is called the spectral radius of G, λ n ( G ) $λ_{n}(G)$ is the least eigenvalue of G, and the spread of G is defined to be the difference between λ 1 ( G ) $λ_{1}(G)$ and λ n ( G ) $λ_{n}(G)$ . 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 $n\ge6$ .