of five spectral moments, we can solve the SDP’s in (\[Bound min eigenval\]) and ([Bound max eigenval]{}) analytically for $s=2$. In this case, the localizing matrix is:$$H_{2}\left( c\right) =\left[ 
\begin{array}{ccc}
m_{1}-c & m_{2}-cm_{1} & m_{3}-cm_{2} \\ 
m_{2}-cm_{1} & m_{3}-cm_{2} & m_{4}-cm_{3} \\ 
m_{3}-cm_{2} & m_{4}-cm_{3} & m_{5}-cm_{4}%
\end{array}%
\right] .  \label{H2c}$$As we proved in Section \[Spectral Section\], the spectral moments in the localizing matrix depend on the number of nodes, edges, cycles of length 3 to 5, the sum-of-squares of degrees $W_{2}$, and the degree-triangle correlation $\mathcal{C}_{dt}$.

Furthermore, for $s=2$, the optimal values $\alpha _{2}$ and $\beta _{2}$ can be analytically computed, as follows. First, note that $%
H_{2}(c)\succcurlyeq 0$ (resp. $-H_{2}(c)\succcurlyeq 0$) if and only if all the eigenvalues of $H_{2}$ are nonnegative (resp. nonpositive). For a given sequence of five moments, the characteristic polynomial of $H_{2}\left(
c\right) $ can be written as$$\phi _{2}\left( \lambda \right) \triangleq \det \left( \lambda
I-H_{2}(c)\right) =\lambda ^{3}+p_{1}\left( c\right) \lambda
^{2}+p_{2}\left( c\right) \lambda +p_{3}\left( c\right) ,$$where $p_{j}\left( c\right) $ is a polynomial of degree $j$ in the variable $%
c$ (with coefficients depending on the moments). Thus, by Descartes’ rule, all the eigenvalues of $H_{2}\left( c\right) $ are nonpositive if and only if $p_{j}\left( 