Table 7.

Matrix construction of graph $G_2$ in the top panel of Figure 5.

$n$	$n=0$	$n=1$	$n=2$
${\Omega }_n$	$\text{span}\left\{e_1,e_2,e_3,e_4,e_5\right\}$	$\text{span}\left\{e_{13},e_{25},e_{32},e_{34},e_{45}\right\}$	{0}
$B_{n+1}$	$\begin{array}{c}\\ e_1\\ e_2\\ e_3\\ e_4\\ e_5\end{array}\begin{array}{c}\begin{array}{ccccc}{e}_{13}& e_{25}& e_{32}& e_{34}& e_{45}\end{array}\\ \left(\begin{array}{ccccc}-1& 0& 0& 0& 0\\ 0& -1& 1& 0& 0\\ 1& 0& -1& -1& 0\\ 0& 0& 0& 1& 1\\ 0& 1& 0& 0& -1\end{array}\right)\end{array}$	$5\times 0$ empty matrix	(/)
$L_n$	$\left(\begin{array}{ccccc}1& 0& -1& 0& 0\\ 0& 2& -1& 0& -1\\ -1& -1& 3& -1& 0\\ 0& 0& -1& 2& -1\\ 0& -1& 0& -1& 2\end{array}\right)$	$\left(\begin{array}{ccccc}2& 0& -1& -1& 0\\ 0& 2& -1& 0& -1\\ -1& -1& 2& 1& 0\\ -1& 0& 1& 2& 1\\ 0& -1& 0& 1& 2\end{array}\right)$	(/)
${\beta }_n$	1	1	0
$\text{Spectra}\left(L_n\right)$	{0, 0.8299, 2, 2.6889, 4.4812}	{0, 0.8299, 2, 2.6889, 4.4812}	/