The Largest Laplacian Eigenvalue Of Unicyclic And Bicyclic Graphs With N Vertices And Fixed Diameter D

The set of unicyclic and bicyclic graphs with n vertices and diameter d is denoted asU (n,d)and D(n,d) respectively.In this paper, we consider the graphs with d≥3 and determine all unicyclic and bi-cyclic graphs in U (n,d) and D(n,d) whose largest Laplacian eigenvalue is the greatestamong all the unicyclic and all the bicyclic graphs with n vertices and fixed diameter, respec-tively.We study the largest Laplacian eigenvalues of unicyclic and bicyclic graphs with nvertices and diameter d. We determined all graphs in D(n,d) whose largest Laplacianeigenvalue is greater than n - d + 2. And among those graphs, we characterize the specificset of graphs whose largest Laplacian eigenvalue is greater then other sets of graphs.For n - d≥12,d≥3 ,we further prove further that there is a unique graph whichlargest Laplacian eigenvalue is the largest in D(n,d)...