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Article Contents

# Persistent path Laplacian

• * Corresponding author: Guo-Wei Wei

This work was supported in part by NIH grants R01GM126189 and R01AI164266, NSF grants DMS-2052983, DMS-1761320, and IIS-1900473, NASA grant 80NSSC21M0023, Michigan Economic Development Corporation, MSU Foundation, Bristol-Myers Squibb 65109, and Pfizer

• Path homology proposed by S.-T.Yau and his co-workers provides a new mathematical model for directed graphs and networks. Persistent path homology (PPH) extends the path homology with filtration to deal with asymmetry structures. However, PPH is constrained to purely topological persistence and cannot track the homotopic shape evolution of data during filtration. To overcome the limitation of PPH, persistent path Laplacian (PPL) is introduced to capture the shape evolution of data. PPL's harmonic spectra fully recover PPH's topological persistence and its non-harmonic spectra reveal the homotopic shape evolution of data during filtration.

Mathematics Subject Classification: Primary: 62R40; Secondary: 55N31.

 Citation:

• Figure 1.  Homologies of directed subgraphs. a, b, and c illustrate three subgraphs whose homology groups or homology group dimensions are related to the original digraphs

Figure 2.  Five digraphs. a and b Digraphs with 3 vertices and 3 directed edges. c and d Digraphs with 4 vertices and 4 directed edges. e A digraph with 6 vertices and 8 directed edges. f A digraph with 6 vertices and 8 directed edges

Figure 3.  Illustration of filtration on a tetrahedron. Here, $1, 2, 3,$ and $4$ represent four elementary $0$-paths $e_1, e_2, e_3,$ and $e_4$. The top panel is a tetrahedron that has edge lengths $|e_{12}| = |e_{32}| = |e_{24}| = 1$ and $|e_{13}| = |e_{14}| = |e_{34}| = \sqrt{2}$. The bottom panel is a tetrahedron that has edge lengths $|e_{32}| = |e_{24}| = 1$, $|e_{34}| = \sqrt{2}$, $|e_{12}| = \sqrt{3}$, and $|e_{13}| = | e_{14}| = 2$

Figure 4.  Comparison of Betti numbers and non-harmonic spectra of $L_n^{\delta, \delta}$ when $n = 0, 1,$ and 2 on tetrahedrons Tetra 1 and Tetra 2. Note that since $\beta_{1}^{\delta, \delta} = 0$ and $\beta_{2}^{\delta, \delta} = 0$ for Tetra 1 and Tetra 2, topological variants from persistent path homology cannot discriminate Tetra 1 and Tetra 2. However $\lambda_{1}^{\delta, \delta}$ and $\lambda_{2}^{\delta, \delta}$ show the differences between Tetra 1 and Tetra 2

Figure 5.  Illustration of filtration on a pyramid. Here, $1, 2, 3, 4,$ and $5$ represent five elementary $0$-paths $e_1, e_2, e_3, e_4,$ and $e_5$. The top panel is a pyramid that has edge lengths $|e_{13}| = | e_{25}| = | e_{32}| = | e_{34}| = | e_{54}| = 1$, $|e_{12}| = |e_{14} | = \sqrt{2}$, and $|e_{15} | = \sqrt{3}$. The bottom panel is a pyramid that has edge lengths $|e_{25}| = |e_{32}| = |e_{34}| = |e_{54}| = 1$, $|e_{12}| = |e_{14}| = 2$, and $|e_{15} | = \sqrt{5}$

Figure 6.  Comparison of Betti number and non-harmonic spectra of $L_n^{\delta, \delta}$ when $n = 0, 1,$c and $2$ on pyramids Pyra 1 and Pyra 2. Note that since $\beta_{2}^{\delta, \delta} = 0$, it cannot distinguish Pyra 1 and Pyra 2. But $\lambda_{2}^{\delta, \delta}$ can tell the difference

Figure 7.  a The 3D structures of CB7, 2 glycolurils, and path direction assignment. Here, from left to right, the side view of CB7, top view of CB7, the structure of two glycoluril units ( = C$_{10}$H$_4$N$_8$O$_4$ = ), and electronegativity-based path direction assignment are depicted as well. b Illustration of filtration-induced geometries $G_i (i = 1, 2, ..., 8)$ of CB7. Eight digraphs $G_1 =$$G_0^{0.200*2}, G_2 = G_0^{0.565*2}, G_3 = G_0^{0.710*2},$$ G_4 = G_0^{0.745*2},$$G_5 = G_0^{0.800*2},$$ G_6 = G_0^{1.210*2},$$G_7 = G_0^{1.315*2},$$G_8 = G_0^{1.800*2}$ are constructed under filtration parameter $\delta$. c Illustration of filtration-induced path complexes within two glycoluril units. Path directions can be inferred from their colors as shown in the last chart of a. d Betti numbers $\beta_n^{\delta, \delta}$ and non-harmonic spectra $\tilde{\lambda}_n^{\delta, \delta}$ of persistent path Laplacians ($L_n^{\delta, \delta}$ when $n = 0, 1,$ and $2$) for CB7

Table 1.  Illustration of digraph c in Figure 2

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_1, e_2, e_3, e_4\}$ $\text{span}\{e_{12}, e_{14}, e_{23}, e_{43}\}$ $\text{span}\{e_{143} - e_{123}\}$ $B_{n+1}$ $\begin{array}{l} \begin{array}{*{20}{c}} & e_{12} & e_{14} & e_{23} & e_{43}\end{array}\\ \begin{array}{*{20}{c}} e_1 \\ e_2 \\ e_3 \\ e_4 \end{array}\left( {\begin{array}{*{20}{c}} -1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & 1 & -1 \end{array}} \right) \end{array}$ $\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {}&{{e_{143}} - {e_{123}}} \end{array}}\\ {\begin{array}{*{20}{c}} {{e_{12}}}\\ {{e_{14}}}\\ {{e_{23}}}\\ {{e_{43}}} \end{array}\left( {\begin{array}{*{20}{c}} { - 1}\\ 1\\ { - 1}\\ 1 \end{array}} \right)} \end{array}$ $1\times 0$ empty matrix $L_n$ $\left(\begin{array}{cccc}2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2\end{array}\right)$ $\left(\begin{array}{cccc}3 & 0 & 0 & -1 \\ 0 & 3 & -1 & 0 \\ 0 & -1 & 3 & 0 \\ -1 & 0 & 0 & 3\end{array}\right)$ (4) $\beta_{n}$ 1 0 0 $\text{Spectra}(L_{n})$ $\{0, 2, 2, 4\}$ $\{2, 2, 4, 4\}$ $\{4\}$

Table 2.  Illustration of digraph d in Figure 2

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_1, e_2, e_3, e_4\}$ $\text{span}\{e_{12}, e_{14}, e_{32}, e_{34}\}$ $\{0\}$ $B_{n+1}$ $\begin{array}{l} \begin{array}{*{20}{c}} & e_{12} & e_{14} & e_{32} & e_{34}\end{array}\\ \begin{array}{*{20}{c}} e_1 \\ e_2 \\ e_3 \\ e_4 \end{array}\left( {\begin{array}{*{20}{c}} -1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & -1 & 1 \end{array}} \right) \end{array}$ $4\times 0$ empty matrix $\left(\begin{array}{ccccc} / \end{array}\right)$ $L_n$ $\left(\begin{array}{cccc}2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2\end{array}\right)$ $\left(\begin{array}{llll}2 & 1 & 1 & 0 \\ 1 & 2 & 0 & 1 \\ 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2\end{array}\right)$ $\left(\begin{array}{ccccc} / \end{array}\right)$ $\beta_{n}$ 1 1 0 $\text{Spectra}(L_{n})$ $\{0, 2, 2, 4\}$ $\{0, 2, 4, 4\}$ /

Table 3.  Illustration of digraph e in Figure 2

Table 4.  Illustration of digraph f in Figure 2

Table 5.  Matrix construction of graph $G_1$ (with isolated points included) in the top panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_1, e_2, e_3, e_4, e_5\}$ $\{0\}$ $\{0\}$ $B_{n+1}$ $5\times 0$ empty matrix / / $L_n$ $5\times 5$ zero matrix / / $\beta_{n}$ 5 / / $\text{Spectra}(L_{n})$ $\{0, 0, 0, 0, 0\}$ / /

Table 6.  Matrix construction of graph $G_1$ (without isolated points) in the top panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\{0\}$ $\{0\}$ $\{0\}$ $B_{n+1}$ / / / $L_n$ / / / $\beta_{n}$ / / / $\text{Spectra}(L_{n})$ / / /

Table 7.  Matrix construction of graph $G_2$ in the top panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_1, e_2, e_3, e_4, e_5\}$ $\text{span}\{e_{13}, e_{25}, e_{32}, e_{34}, e_{45}\}$ $\{0\}$ $B_{n+1}$ $\begin{array}{@{}r@{}c@{}c@{}c@{}c@{}c@{}l@{}} & e_{13} & e_{25} & e_{32} & e_{34} & e_{45} \\ \left.\begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e_4 \\ e_5 \end{array}\right( & \begin{array}{c}-1 \\ 0 \\ 1 \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ -1 \\ 0 \\ 0 \\ 1 \end{array} & \begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ -1 \\ 1 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\ -1 \end{array} & \left)\begin{array}{c} \\ \\ \\ \\ \\ \end{array}\right. \end{array}$ $5\times 0$ empty matrix $\left(\begin{array}{ccccc} / \end{array}\right)$ $L_n$ $\left(\begin{array}{ccccc} 1 & 0 & -1 & 0 & 0 \\ 0 & 2 & -1 & 0 & -1 \\ -1 & -1 & 3 & -1 & 0 \\ 0 & 0 & -1 & 2 & -1 \\ 0 & -1 & 0 & -1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} 2 & 0 & -1 & -1 & 0 \\ 0 & 2 & -1 & 0 & -1 \\ -1 & -1 & 2 & 1 & 0 \\ -1 & 0 & 1 & 2 & 1 \\ 0 & -1 & 0 & 1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} / \end{array}\right)$ $\beta_{n}$ 1 1 0 $\text{Spectra}(L_{n})$ $\{0, 0.8299, 2, 2.6889, 4.4812\}$ $\{0, 0.8299, 2, 2.6889, 4.4812\}$ /

Table 8.  Matrix construction of graph $G_3$ in the top panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_1, e_2, e_3, e_4, e_5\}$ $\text{span}\{e_{12}, e_{13}, e_{14}, e_{25}, e_{32}, e_{34}, e_{54}\}$ $\text{span}\{e_{132}, e_{134}\}$ $B_{n+1}$ $\begin{array}{c} & e_{12} & e_{13} & e_{14} & e_{25} & e_{32} & e_{34} & e_{54} \\ \left.\begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e_4 \\ e_5 \end{array}\right( & \begin{array}{c} -1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} & \begin{array}{c} -1 \\ 0 \\ 1 \\ 0 \\ 0 \end{array} & \begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array} & \begin{array}{c} 0 \\ -1 \\ 0 \\ 0 \\ 1 \end{array} & \begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ -1 \\ 1 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\-1 \end{array} & \left)\begin{array}{c} \\ \\ \\ \\ \\ \end{array}\right. \end{array}$ $\begin{array}{c} & e_{132} & e_{134} \\ \left.\begin{array}{c} e_{12} \\ e_{13} \\ e_{14} \\ e_{25} \\ e_{32} \\ e_{34} \\ e_{54} \end{array}\right( & \begin{array}{c} -1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array} & \left)\begin{array}{c} \\ \\ \\ \\ \\ \\ \\ \end{array}\right. \end{array}$ $2\times 0$ empty matrix $L_n$ $\left(\begin{array}{ccccc} 3 & -1 & -1 & -1 & 0 \\ -1 & 3 & -1 & 0 & -1 \\ -1 & -1 & 3 & -1 & 0 \\ -1 & 0 & -1 & 3 & -1 \\ 0 & -1 & 0 & -1 & 2 \end{array}\right)$ $\left(\begin{array}{cccccccccc} 3 & 0 & 1 & -1 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 3 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 2 & -1 & 0 & -1 \\ 0 & 0 & 0 & -1 & 3 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 3 & 1 \\ 0 & 0 & 1 & -1 & 0 & 1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} 3 & 1 \\ 1 & 3 \end{array}\right)$ $\beta_{n}$ 1 1 0 $\text{Spectra}(L_{n})$ $\{0, 2, 3, 4, 5\}$ $\{0, 2, 2, 3, 4, 4, 5\}$ $\{2, 4\}$

Table 9.  Matrix construction of graph $G_4$ in the top panel of Figure 5

Table 10.  Matrix construction of graph $G_5$ in the top panel of Figure 5

Table 11.  Matrix construction of graph $G_1$ (with isolated points included) in the bottom panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_1, e_2, e_3, e_4, e_5\}$ / / $B_{n+1}$ $5\times 0$ empty matrix / / $L_n$ $5\times 5$ zero matrix / / $\beta_{n}$ 5 / / $\text{Spectra}(L_{n})$ $\{0, 0, 0, 0, 0\}$ / /

Table 12.  Matrix construction of graph $G_1$ (without isolated points) in the bottom panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\{0\}$ $\{0\}$ $\{0\}$ $B_{n+1}$ / / / $L_n$ / / / $\beta_{n}$ / / / $\text{Spectra}(L_{n})$ / / /

Table 13.  Matrix construction of graph $G_2$ (with isolated points included) in the bottom panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_1, e_2, e_3, e_4, e_5\}$ $\text{span}\{e_{25}, e_{32}, e_{34}, e_{54}\}$ $\{0\}$ $B_{n+1}$ $\begin{array}{c} \begin{array}{c} & e_{25} & e_{32} & e_{34} & e_{54} \end{array} \\ \left.\begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e_4 \\ e_5 \end{array}\right. \left(\begin{array}{ccccc} & \begin{array}{c} 0 \\ -1 \\ 0 \\ 0 \\ 1 \end{array} & \begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ -1 \\ 1 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\ -1 \end{array} \end{array}\right) \end{array}$ $4\times 0$ empty matrix $\left(\begin{array}{ccccc} / \end{array}\right)$ $L_n$ $\left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & -2 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 2 & 1 \\ 0 & -2 & 0 & 1 & 3 \end{array}\right)$ $\left(\begin{array}{ccccc} 2 & 0 & 1 & -2 \\ 0 & 2 & -1 & 0 \\ 1 & -1 & 2 & -1 \\ -2 & 0 & -1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} / \end{array}\right)$ $\beta_{n}$ 2 1 0 $\text{Spectra}(L_{n})$ $\{0, 0, 0.6571, 2.5293, 4.8136\}$ $\{0, 0.6571, 2.5293, 4.8136\}$ /

Table 14.  Matrix construction of graph $G_2$ (without isolated points) in the bottom panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_2, e_3, e_4, e_5\}$ $\text{span}\{e_{25}, e_{32}, e_{34}, e_{54}\}$ $\{0\}$ $B_{n+1}$ $\begin{array}{c} & e_{25} & e_{32} & e_{34} & e_{54} \\ \left.\begin{array}{c} e_2 \\ e_3 \\ e_4 \\ e_5 \end{array}\right( & \begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array} & \begin{array}{c} 1 \\ -1 \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ -1 \\ 1 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ 1 \\ -1 \end{array} & \left)\begin{array}{c} \\ \\ \\ \\ \end{array}\right. \end{array}$ $4\times 0$ empty matrix $\left(\begin{array}{ccccc} / \end{array}\right)$ $L_n$ $\left(\begin{array}{ccccc} 2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} 2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & 1 & 2 & 1 \\ -1 & 0 & 1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} / \end{array}\right)$ $\beta_{n}$ 1 1 0 $\text{Spectra}(L_{n})$ $\{0, 2, 2, 4\}$ $\{0, 2, 2, 4\}$ /

Table 15.  Matrix construction of graph $G_3$ (with isolated points included) in the bottom panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_1, e_2, e_3, e_4, e_5\}$ $\text{span}\{e_{25}, e_{32}, e_{34}, e_{54}\}$ $\{0\}$ $B_{n+1}$ $\begin{array}{c} & e_{25} & e_{32} & e_{34} & e_{54} \\ \left.\begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e_4 \\ e_5 \end{array}\right( & \begin{array}{c} 0 \\ -1 \\ 0 \\ 0 \\ 1 \end{array} & \begin{array}{c} 0 \\ 1 \\ -1 \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ -1 \\ 1 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\ -1 \end{array} & \left)\begin{array}{c} \\ \\ \\ \\ \\ \end{array}\right. \end{array}$ $4\times 0$ empty matrix $\left(\begin{array}{ccccc} / \end{array}\right)$ $L_n$ $\left(\begin{array}{ccccc} 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & -2 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 2 & 1 \\ 0 & -2 & 0 & 1 & 3 \end{array}\right)$ $\left(\begin{array}{ccccc} 2 & 0 & 1 & -2 \\ 0 & 2 & -1 & 0 \\ 1 & -1 & 2 & -1 \\ -2 & 0 & -1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} / \end{array}\right)$ $\beta_{n}$ 2 1 0 $\text{Spectra}(L_{n})$ $\{0, 0, 0.6571, 2.5293, 4.8136\}$ $\{0, 0.6571, 2.5293, 4.8136\}$ /

Table 16.  Matrix construction of graph $G_3$ (without isolated points) in the bottom panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_2, e_3, e_4, e_5\}$ $\text{span}\{e_{25}, e_{32}, e_{34}, e_{54}\}$ $\{0\}$ $B_{n+1}$ $\begin{array}{c} & e_{25} & e_{32} & e_{34} & e_{54} \\ \left.\begin{array}{c} e_2 \\ e_3 \\ e_4 \\ e_5 \end{array}\right( & \begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array} & \begin{array}{c} 1 \\ -1 \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ -1 \\ 1 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ 1 \\ -1 \end{array} & \left)\begin{array}{c} \\ \\ \\ \\ \end{array}\right. \end{array}$ $4\times 0$ empty matrix $\left(\begin{array}{ccccc} / \end{array}\right)$ $L_n$ $\left(\begin{array}{ccccc} 2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} 2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & 1 & 2 & 1 \\ -1 & 0 & 1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} / \end{array}\right)$ $\beta_{n}$ 1 1 0 $\text{Spectra}(L_{n})$ $\{0, 2, 2, 4\}$ $\{0, 2, 2, 4\}$ /

Table 17.  Matrix construction of graph $G_2$ in the bottom panel of Figure 5

 $n$ $n=0$ $n=1$ $n=2$ $\Omega_n$ $\text{span}\{e_1, e_2, e_3, e_4\}$ $\text{span}\{e_{12}, e_{14}, e_{32}, e_{34}\}$ $\{0\}$ $B_{n+1}$ $\begin{array}{c} & e_{12} & e_{14} & e_{32} & e_{34} & e_{54} \\ \left.\begin{array}{c} e_1 \\ e_2 \\ e_3 \\ e_4 \\ e_5 \end{array}\right( & \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} & \begin{array}{c} 0 \\ -1 \\ 0 \\ 0 \\ 1 \end{array} & \begin{array}{c} 0 \\ 0 \\ -1 \\ -1 \\ 0 \end{array} & \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \\ 1 \end{array} & \begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \\ -1 \end{array} & \left)\begin{array}{c} \\ \\ \\ \\ \\ \end{array}\right. \end{array}$ $4\times 0$ empty matrix $\left(\begin{array}{ccccc} / \end{array}\right)$ $L_n$ $\left(\begin{array}{ccccc} 2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} 2 & 1 & 1 & 0 \\ 1 & 2 & 0 & 1 \\ 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 2 \end{array}\right)$ $\left(\begin{array}{ccccc} / \end{array}\right)$ $\beta_{n}$ 1 1 0 $\text{Spectra}(L_{n})$ $\{0, 2, 2, 4\}$ $\{0, 2, 4, 4\}$ /

Table 18.  Matrix construction of graph $G_4$ in the bottom panel of Figure 5

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

Table 19.  Matrix construction of graph $G_5$ in the bottom panel of Figure 5

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