of length $k$ in $\mathcal{G}$ [@Big93]:

\[Moments from Walks\]Let $\mathcal{G}$ be a simple graph. The $k$-th spectral moment of the adjacency matrix of $\mathcal{G}$ can be written as$$m_{k}(A_{\mathcal{G}})=\frac{1}{n}\sum_{i=1}^{n}\lambda_{i}^{k}=\frac{1}{n}%
\left\vert \Psi_{\mathcal{G}}^{\left( k\right) }\right\vert ,
\label{Moments as Walks in Graph}$$ where $\Psi_{\mathcal{G}}^{\left( k\right) }$ is the set of all closed walks of length $k$ in $\mathcal{G}$. [^5]

From (\[Moments as Walks in Graph\]), we can easily compute the first three spectral moments of $A_{\mathcal{G}}$ in terms of the number of nodes, edges and triangles as follows [@Big93]:

\[One corollary\]Let $\mathcal{G}$ be a simple graph with adjacency matrix $A_{\mathcal{G}}$. Denote by $n$, $e$ and $\Delta $ the number of nodes, edges and triangles in $\mathcal{G}$, respectively. Then,$$\begin{aligned}
m_{1}(A_{\mathcal{G}})& =0,  \label{Moments as Averages} \\
m_{2}(A_{\mathcal{G}})& =2e/n,  \notag \\
m_{3}(A_{\mathcal{G}})& =6\,\Delta /n.  \notag\end{aligned}$$

Notice that the coefficients 2 (resp. 6) in the above expressions corresponds to the number of closed walks of length 2 (resp. 3) enabled by the presence of an edge (resp. triangle). Similar expressions can be derived for higher-order spectral moments, although a more elaborated combinatorial analysis in required.

In our case, we are interested in the following expressions, derived in [PJ10]{}, for the first five spectral moments of $\mathcal{G}$:

\[Lemma 4th 