and pentagons $%
p_{i}$ touching each node $i\in \mathcal{V}\left( \mathcal{F}\right) $. In Fig. 1, we plot the distributions of degrees and triangles, as well as a scatter plot of $t_{i}$ versus $d_{i}$ (where each point has coordinates $%
\left( d_{i},t_{i}\right) $, in log-log scale, for all $i\in \mathcal{V}%
\left( \mathcal{F}\right) $). We can aggregate those quantities that are relevant to compute the spectral moments to obtain the following numerical values:$$\begin{array}{ll}
e/n=\sum d_{i}/2n=9.478, & \Delta /n=\sum t_{i}/3n=28.15, \\ 
Q/n=\sum q_{i}/4n=825.3, & \Pi /n=\sum p_{i}/5n=31,794, \\ 
W_{2}/n=\sum d_{i}^{2}/n=1,318, & \mathcal{C}_{dt}/n=\sum d_{i}t_{i}/n=8,520.%
\end{array}%$$Hence, using Corollary \[One corollary\] and Lemma \[Lemma 4th metrics\], we obtain the following values for the spectral moments: $m_{1}\left( A_{%
\mathcal{F}}\right) =0,$ $m_{2}\left( A_{\mathcal{F}}\right) =18.95,$ $%
m_{3}\left( A_{\mathcal{F}}\right) =168.9,$ $m_{4}\left( A_{\mathcal{F}%
}\right) =9,230,$ and $m_{5}\left( A_{\mathcal{F}}\right) =402,310.$

In this section, we have derived expressions to compute the first five spectral moment of $A_{\mathcal{G}}$ from network structural properties. In the next section, we use semidefinite programming to extract information regarding eigenvalues of interest from a sequence of spectral moments.

\[Optimal Bounds\]Optimal Spectral Bounds from Spectral Moments
===============================================================

Here, we introduce an approach to derive an upper bound on the smallest eigenvalues of $A_{\mathcal{G}}$ from its sequence of spectral moments[^6]. Since we 