On the spectral moment of graphs with $k$ cut edges

Let $A(G)$ be the adjacency matrix of a graph $G$ with ${\lambda }_1(G)$, ${\lambda }_2(G)$, …, ${\lambda }_n(G)$ being its eigenvalues in non-increasing order. Call the number $S_k(G):={\sum }_{i=1}^n{\lambda }_i^k(G)(k=0,1,\mathrm{.}\mathrm{.}\mathrm{.},n-1)$ the $k$th spectral moment of $G$. Let $S(G)=(S_0(G),S_1(G),\mathrm{.}\mathrm{.}\mathrm{.},S_{n-1}(G))$ be the sequence of spectral moments of $G$. For two graphs $G_1$ and $G_2$, we have $G_1{\prec }_s{G}_2$ if $S_i(G_1)=S_i(G_2)(i=0,1,\mathrm{.}\mathrm{.}\mathrm{.},k-1)$ and $S_k(G_1)<S_k(G_2)$ for some $k\in 1,2,\mathrm{.}\mathrm{.}\mathrm{.},n-1$. Denote by ${\mathcal{G}}_n^k$ the set of connected $n$-vertex graphs with $k$ cut edges. In this paper, we determine the first, the second, the last and the second last graphs, in an $S$-order, among ${\mathcal{G}}_n^k$, respectively.