The largest eigenvalue of unicyclic graphs
The author shows that the largest eigenvalue of the adjacency matrix of a unicyclic graph with the maximum vertex degree \(\Delta\) is bounded from above by \(2\sqrt{\Delta-1}\), while the largest eigenvalue of its Laplacian matrix is bounded by \(\Delta+2\sqrt{\Delta-1}\), with equality in the first case holding for all cycles, and in the second case for even cycles. These results improve previous results of \textit{C. D. Godsil} [Ann. Discrete Math. 20, 151-159 (1984; Zbl 0559.05040)] and of the reviewer [Linear Algebra Appl. 360, 35-42 (2003; Zbl 1028.05062)] on the largest eigenvalue of trees with fixed maximum degree.
scientific article; zbMATH DE number 5124995