On the spectral moments of unicyclic graphs with fixed diameter

Unicyclic graphs are connected graphs in which the number of edges equals the number of vertices. Let Un,d be the class of unicyclic graphs of order n and diameter d. For unicyclic graphs, lexicographic ordering by spectral moments (S-order) is discussed in this paper. The last d+⌊d2⌋-2 graphs, in the S-order, among all unicyclic graphs in Un,d(3⩽d⩽n-5) are characterized. In addition, for the cases d=n-4 and d=n-3, the last 2n-11 and last n+⌊n2⌋-7 unicyclic graphs in the set Un,d are also given, respectively.