Graphs with the same CDC on 8 Vertices

Here we give an exhaustive list of all 32 pairs of non-isomorphic graphs on 8 vertices which have the same CDC. This automatically includes those graphs on ≤ 8 vertices since any \(n\leq8\) vertex pair of graphs \(G\), \(H\) with the same CDC will appear here as the pair \(G+(8-n)K_1\), \(H+(8-n)K_1\). For relevant definitions, please refer to my undergraduate dissertation, or my recent publication with Irene Sciriha.

The numbering of the graphs here is in accordance with the graph data on Brendan McKay's webpage. The colouring of the vertices (and walk matrix rows) is so that vertices of the same colour have the same number of \(k\)-walks, for any value of \(k\).

It's easy to work with these graphs in Mathematica by importing them from McKay's webpage; the pairs of numbers are available as CSV files from the root directory of this site in the form cdcX.csv. (X here stands for the number of vertices; the graphs pairs on this webpage correspond to those in cdc8.csv). For instance:

Click to copy
CDC Graph Data Example

I have also computed the pairs for X=6, 7 and 9, you can see more about those here. Notice that I am making use of the Walks package above.

Ryser Switches

A Ryser switch is a degree-sequence preserving operation on a pair of edges in a given graph. In the list below, it turns out that all but one pair of graphs is the result of a single Ryser switch. (The exceptional pair (5358, 11716) requires two).

Given a graph \(G\) and edges \(uv, xy\in E(G)\) such that \(ux, vy\notin E(G)\), the Ryser switch on the edges \(uv\) and \(xy\) replaces those edges with \(ux\) and \(vy\). We denote the graph obtained by \(\operatorname{ryser}(G,uv,xy)\).


Pair (447, 958)


Graph 447

Graph 958
\(\begin{pmatrix} \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{blue}1&\color{blue}0\\ \color{blue}1&\color{blue}0\\ \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{red}1&\color{red}2 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 447: \(-1\), \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{0}\), \(\boldsymbol{0}\), \(\boldsymbol{2}\)
Eigenvalues of 958: \(-1\), \(-1\), \(-1\), \(-1\), \(\boldsymbol{0}\), \(\boldsymbol{0}\), \(\boldsymbol{2}\), \(\boldsymbol{2}\)

CDC

Ryser Switch
\(G_{958}=\operatorname{ryser}(G_{447},16, 38)\)

Pair (1030, 1162)


Graph 1030

Graph 1162
\(\begin{pmatrix} \color{red}1&\color{red}2&\color{red}5\\ \color{red}1&\color{red}2&\color{red}5\\ \color{blue}1&\color{blue}1&\color{blue}3\\ \color{blue}1&\color{blue}1&\color{blue}3\\ \color{red}1&\color{red}2&\color{red}5\\ \color{red}1&\color{red}2&\color{red}5\\ \color{green}1&\color{green}3&\color{green}5\\ \color{green}1&\color{green}3&\color{green}5 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 1030: \(-1\), \(-1\), \(\boldsymbol{0.31}\), \(\boldsymbol{0.31}\), \(\boldsymbol{-1.48}\), \(\boldsymbol{-1.48}\), \(\boldsymbol{2.17}\), \(\boldsymbol{2.17}\)
Eigenvalues of 1162: \(-0.31\), \(-1\), \(1\), \(1.48\), \(-2.17\), \(\boldsymbol{0.31}\), \(\boldsymbol{-1.48}\), \(\boldsymbol{2.17}\)

CDC

Ryser Switch
\(G_{1162}=\operatorname{ryser}(G_{1030},15, 26)\)

Pair (1032, 1164)


Graph 1032

Graph 1164
\(\begin{pmatrix} \color{red}1&\color{red}2&\color{red}5&\color{red}11\\ \color{red}1&\color{red}2&\color{red}5&\color{red}11\\ \color{blue}1&\color{blue}2&\color{blue}6&\color{blue}12\\ \color{green}1&\color{green}0&\color{green}0&\color{green}0\\ \color{red}1&\color{red}2&\color{red}5&\color{red}11\\ \color{red}1&\color{red}2&\color{red}5&\color{red}11\\ \color{yellow}1&\color{yellow}3&\color{yellow}6&\color{yellow}16\\ \color{yellow}1&\color{yellow}3&\color{yellow}6&\color{yellow}16 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 1032: \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{0}\), \(\boldsymbol{0.47}\), \(\boldsymbol{-1.81}\), \(\boldsymbol{2.34}\)
Eigenvalues of 1164: \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{0}\), \(\boldsymbol{0.47}\), \(\boldsymbol{-1.81}\), \(\boldsymbol{2.34}\)

CDC

Ryser Switch
\(G_{1164}=\operatorname{ryser}(G_{1032},15, 26)\)

Pair (1105, 1235)


Graph 1105

Graph 1235
\(\begin{pmatrix} \color{red}1&\color{red}2&\color{red}6\\ \color{red}1&\color{red}2&\color{red}6\\ \color{blue}1&\color{blue}2&\color{blue}8\\ \color{blue}1&\color{blue}2&\color{blue}8\\ \color{red}1&\color{red}2&\color{red}6\\ \color{red}1&\color{red}2&\color{red}6\\ \color{green}1&\color{green}4&\color{green}8\\ \color{green}1&\color{green}4&\color{green}8 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 1105: \(0\), \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{0.64}\), \(\boldsymbol{-2.32}\), \(\boldsymbol{2.68}\)
Eigenvalues of 1235: \(0\), \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{0.64}\), \(\boldsymbol{-2.32}\), \(\boldsymbol{2.68}\)

CDC

Ryser Switch
\(G_{1235}=\operatorname{ryser}(G_{1105},15, 26)\)

Pair (3392, 3494)


Graph 3392

Graph 3494
\(\begin{pmatrix} \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{blue}1&\color{blue}3\\ \color{blue}1&\color{blue}3\\ \color{blue}1&\color{blue}3\\ \color{blue}1&\color{blue}3 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 3392: \(0\), \(0\), \(-1\), \(1\), \(\boldsymbol{\frac{1}{2} \left(1-\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(1+\sqrt{17}\right)}\), \(\frac{1}{2} \left(-1-\sqrt{17}\right)\), \(\frac{1}{2} \left(\sqrt{17}-1\right)\)
Eigenvalues of 3494: \(0\), \(0\), \(-1\), \(-1\), \(\boldsymbol{\frac{1}{2} \left(1-\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(1-\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(1+\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(1+\sqrt{17}\right)}\)

CDC

Ryser Switch
\(G_{3494}=\operatorname{ryser}(G_{3392},57, 68)\)

Pair (3482, 3592)


Graph 3482

Graph 3592
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}11\\ \color{blue}1&\color{blue}2&\color{blue}7\\ \color{red}1&\color{red}3&\color{red}11\\ \color{blue}1&\color{blue}2&\color{blue}7\\ \color{green}1&\color{green}3&\color{green}9\\ \color{green}1&\color{green}3&\color{green}9\\ \color{yellow}1&\color{yellow}4&\color{yellow}11\\ \color{yellow}1&\color{yellow}4&\color{yellow}11 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 3482: \(-0.31\), \(-1\), \(1\), \(1.48\), \(-2.17\), \(\boldsymbol{0.14}\), \(\boldsymbol{-2.27}\), \(\boldsymbol{3.13}\)
Eigenvalues of 3592: \(0.31\), \(-1\), \(-1\), \(-1.48\), \(2.17\), \(\boldsymbol{0.14}\), \(\boldsymbol{-2.27}\), \(\boldsymbol{3.13}\)

CDC

Ryser Switch
\(G_{3592}=\operatorname{ryser}(G_{3482},25, 46)\)

Pair (3413, 3779)


Graph 3413

Graph 3779
\(\begin{pmatrix} \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{blue}1&\color{blue}3\\ \color{blue}1&\color{blue}3\\ \color{blue}1&\color{blue}3\\ \color{blue}1&\color{blue}3 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 3413: \(0\), \(-1\), \(-1\), \(1\), \(-2\), \(2\), \(\boldsymbol{\frac{1}{2} \left(1-\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(1+\sqrt{17}\right)}\)
Eigenvalues of 3779: \(0\), \(1\), \(1\), \(1\), \(-2\), \(-2\), \(\boldsymbol{\frac{1}{2} \left(1-\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(1+\sqrt{17}\right)}\)

CDC

Ryser Switch
\(G_{3779}=\operatorname{ryser}(G_{3413},15, 48)\)

Pair (1270, 5629)


Graph 1270

Graph 5629
\(\begin{pmatrix} \color{red}1&\color{red}1\\ \color{blue}1&\color{blue}2\\ \color{blue}1&\color{blue}2\\ \color{blue}1&\color{blue}2\\ \color{red}1&\color{red}1\\ \color{blue}1&\color{blue}2\\ \color{blue}1&\color{blue}2\\ \color{blue}1&\color{blue}2 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 1270: \(-1\), \(-1\), \(-1\), \(-2\), \(\boldsymbol{1}\), \(\boldsymbol{1}\), \(\boldsymbol{1}\), \(\boldsymbol{2}\)
Eigenvalues of 5629: \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(\boldsymbol{1}\), \(\boldsymbol{2}\), \(\boldsymbol{2}\)

CDC

Ryser Switch
\(G_{5629}=\operatorname{ryser}(G_{1270},26, 48)\)

Pair (5684, 5718)


Graph 5684

Graph 5718
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}11\\ \color{blue}1&\color{blue}2&\color{blue}6\\ \color{blue}1&\color{blue}2&\color{blue}6\\ \color{red}1&\color{red}3&\color{red}11\\ \color{blue}1&\color{blue}2&\color{blue}6\\ \color{blue}1&\color{blue}2&\color{blue}6\\ \color{green}1&\color{green}4&\color{green}10\\ \color{green}1&\color{green}4&\color{green}10 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 5684: \(-1\), \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{1}\), \(\boldsymbol{-2}\), \(\boldsymbol{3}\)
Eigenvalues of 5718: \(-1\), \(-1\), \(\boldsymbol{1}\), \(\boldsymbol{1}\), \(\boldsymbol{1}\), \(\boldsymbol{-2}\), \(\boldsymbol{-2}\), \(\boldsymbol{3}\)

CDC

Ryser Switch
\(G_{5718}=\operatorname{ryser}(G_{5684},25, 36)\)

Pair (3887, 5753)


Graph 3887

Graph 5753
\(\begin{pmatrix} \color{red}1&\color{red}2\\ \color{blue}1&\color{blue}3\\ \color{blue}1&\color{blue}3\\ \color{green}1&\color{green}0\\ \color{blue}1&\color{blue}3\\ \color{blue}1&\color{blue}3\\ \color{red}1&\color{red}2\\ \color{yellow}1&\color{yellow}4 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 3887: \(-1\), \(1\), \(1\), \(-2\), \(-2\), \(\boldsymbol{0}\), \(\boldsymbol{0}\), \(\boldsymbol{3}\)
Eigenvalues of 5753: \(-1\), \(-1\), \(-1\), \(-2\), \(2\), \(\boldsymbol{0}\), \(\boldsymbol{0}\), \(\boldsymbol{3}\)

CDC

Ryser Switch
\(G_{5753}=\operatorname{ryser}(G_{3887},15, 37)\)

Pair (3899, 5755)


Graph 3899

Graph 5755
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}12\\ \color{red}1&\color{red}3&\color{red}12\\ \color{red}1&\color{red}3&\color{red}12\\ \color{blue}1&\color{blue}0&\color{blue}0\\ \color{red}1&\color{red}3&\color{red}12\\ \color{red}1&\color{red}3&\color{red}12\\ \color{red}1&\color{red}3&\color{red}12\\ \color{green}1&\color{green}6&\color{green}18 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 3899: \(-1\), \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{0}\), \(\boldsymbol{1-\sqrt{7}}\), \(\boldsymbol{1+\sqrt{7}}\)
Eigenvalues of 5755: \(-1\), \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{0}\), \(\boldsymbol{1-\sqrt{7}}\), \(\boldsymbol{1+\sqrt{7}}\)

CDC

Ryser Switch
\(G_{5755}=\operatorname{ryser}(G_{3899},15, 37)\)

Pair (3886, 5756)


Graph 3886

Graph 5756
\(\begin{pmatrix} \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{blue}1&\color{blue}3\\ \color{green}1&\color{green}1\\ \color{blue}1&\color{blue}3\\ \color{red}1&\color{red}2\\ \color{red}1&\color{red}2\\ \color{blue}1&\color{blue}3 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 3886: \(-1\), \(1\), \(1\), \(1\), \(-2\), \(-2\), \(\boldsymbol{1-\sqrt{2}}\), \(\boldsymbol{1+\sqrt{2}}\)
Eigenvalues of 5756: \(-1\), \(-1\), \(-1\), \(1\), \(-2\), \(2\), \(\boldsymbol{1-\sqrt{2}}\), \(\boldsymbol{1+\sqrt{2}}\)

CDC

Ryser Switch
\(G_{5756}=\operatorname{ryser}(G_{3886},15, 37)\)

Pair (3888, 5759)


Graph 3888

Graph 5759
\(\begin{pmatrix} \color{red}1&\color{red}2&\color{red}6&\color{red}20\\ \color{blue}1&\color{blue}3&\color{blue}10&\color{blue}29\\ \color{blue}1&\color{blue}3&\color{blue}10&\color{blue}29\\ \color{green}1&\color{green}1&\color{green}5&\color{green}13\\ \color{blue}1&\color{blue}3&\color{blue}10&\color{blue}29\\ \color{blue}1&\color{blue}3&\color{blue}10&\color{blue}29\\ \color{red}1&\color{red}2&\color{red}6&\color{red}20\\ \color{yellow}1&\color{yellow}5&\color{yellow}13&\color{yellow}45 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 3888: \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{-0.49}\), \(\boldsymbol{0.60}\), \(\boldsymbol{-2.20}\), \(\boldsymbol{3.09}\)
Eigenvalues of 5759: \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{-0.49}\), \(\boldsymbol{0.60}\), \(\boldsymbol{-2.20}\), \(\boldsymbol{3.09}\)

CDC

Ryser Switch
\(G_{5759}=\operatorname{ryser}(G_{3888},15, 37)\)

Pair (3903, 5761)


Graph 3903

Graph 5761
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}13\\ \color{red}1&\color{red}3&\color{red}13\\ \color{red}1&\color{red}3&\color{red}13\\ \color{blue}1&\color{blue}1&\color{blue}7\\ \color{red}1&\color{red}3&\color{red}13\\ \color{red}1&\color{red}3&\color{red}13\\ \color{red}1&\color{red}3&\color{red}13\\ \color{green}1&\color{green}7&\color{green}19 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 3903: \(-1\), \(-1\), \(1\), \(1\), \(\boldsymbol{-2}\), \(\boldsymbol{-2}\), \(\boldsymbol{2-\sqrt{3}}\), \(\boldsymbol{2+\sqrt{3}}\)
Eigenvalues of 5761: \(-1\), \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{-2}\), \(\boldsymbol{2-\sqrt{3}}\), \(\boldsymbol{2+\sqrt{3}}\)

CDC

Ryser Switch
\(G_{5761}=\operatorname{ryser}(G_{3903},15, 37)\)

Pair (6471, 7012)


Graph 6471

Graph 7012
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}10&\color{red}32&\color{red}102\\ \color{red}1&\color{red}3&\color{red}10&\color{red}32&\color{red}102\\ \color{blue}1&\color{blue}2&\color{blue}8&\color{blue}22&\color{blue}78\\ \color{green}1&\color{green}2&\color{green}7&\color{green}21&\color{green}71\\ \color{green}1&\color{green}2&\color{green}7&\color{green}21&\color{green}71\\ \color{yellow}1&\color{yellow}4&\color{yellow}11&\color{yellow}39&\color{yellow}117\\ \color{yellow}1&\color{yellow}4&\color{yellow}11&\color{yellow}39&\color{yellow}117\\ \color{orange}1&\color{orange}4&\color{orange}14&\color{orange}42&\color{orange}142 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 6471: \(-1\), \(-1\), \(2\), \(\boldsymbol{0}\), \(\boldsymbol{0.55}\), \(\boldsymbol{-1.48}\), \(\boldsymbol{-2.29}\), \(\boldsymbol{3.21}\)
Eigenvalues of 7012: \(1\), \(1\), \(-2\), \(\boldsymbol{0}\), \(\boldsymbol{0.55}\), \(\boldsymbol{-1.48}\), \(\boldsymbol{-2.29}\), \(\boldsymbol{3.21}\)

CDC

Ryser Switch
\(G_{7012}=\operatorname{ryser}(G_{6471},14, 25)\)

Pair (6479, 7013)


Graph 6479

Graph 7013
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}8\\ \color{red}1&\color{red}3&\color{red}8\\ \color{blue}1&\color{blue}3&\color{blue}9\\ \color{green}1&\color{green}2&\color{green}6\\ \color{green}1&\color{green}2&\color{green}6\\ \color{red}1&\color{red}3&\color{red}8\\ \color{red}1&\color{red}3&\color{red}8\\ \color{blue}1&\color{blue}3&\color{blue}9 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 6479: \(-1\), \(-1\), \(-1-\sqrt{2}\), \(2\), \(\boldsymbol{0.53}\), \(\boldsymbol{-1.34}\), \(\boldsymbol{2.81}\), \(\sqrt{2}-1\)
Eigenvalues of 7013: \(1\), \(1\), \(-1-\sqrt{2}\), \(-2\), \(\boldsymbol{0.53}\), \(\boldsymbol{-1.34}\), \(\boldsymbol{2.81}\), \(\sqrt{2}-1\)

CDC

Ryser Switch
\(G_{7013}=\operatorname{ryser}(G_{6479},14, 25)\)

Pair (6481, 7015)


Graph 6481

Graph 7015
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}11&\color{red}37\\ \color{red}1&\color{red}3&\color{red}11&\color{red}37\\ \color{blue}1&\color{blue}3&\color{blue}13&\color{blue}43\\ \color{green}1&\color{green}2&\color{green}7&\color{green}24\\ \color{green}1&\color{green}2&\color{green}7&\color{green}24\\ \color{yellow}1&\color{yellow}4&\color{yellow}13&\color{yellow}48\\ \color{yellow}1&\color{yellow}4&\color{yellow}13&\color{yellow}48\\ \color{orange}1&\color{orange}5&\color{orange}17&\color{orange}61 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 6481: \(-1\), \(-1\), \(-2\), \(2\), \(\boldsymbol{-0.28}\), \(\boldsymbol{0.56}\), \(\boldsymbol{-1.79}\), \(\boldsymbol{3.52}\)
Eigenvalues of 7015: \(1\), \(1\), \(-2\), \(-2\), \(\boldsymbol{-0.28}\), \(\boldsymbol{0.56}\), \(\boldsymbol{-1.79}\), \(\boldsymbol{3.52}\)

CDC

Ryser Switch
\(G_{7015}=\operatorname{ryser}(G_{6481},14, 25)\)

Pair (6476, 7026)


Graph 6476

Graph 7026
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}10\\ \color{red}1&\color{red}3&\color{red}10\\ \color{blue}1&\color{blue}2&\color{blue}6\\ \color{red}1&\color{red}3&\color{red}10\\ \color{red}1&\color{red}3&\color{red}10\\ \color{green}1&\color{green}3&\color{green}8\\ \color{green}1&\color{green}3&\color{green}8\\ \color{yellow}1&\color{yellow}4&\color{yellow}12 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 6476: \(-1\), \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{1.15}\), \(\boldsymbol{-2.25}\), \(\boldsymbol{3.10}\)
Eigenvalues of 7026: \(-1\), \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{1.15}\), \(\boldsymbol{-2.25}\), \(\boldsymbol{3.10}\)

CDC

Ryser Switch
\(G_{7026}=\operatorname{ryser}(G_{6476},14, 25)\)

Pair (6478, 7028)


Graph 6478

Graph 7028
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}13\\ \color{red}1&\color{red}3&\color{red}13\\ \color{blue}1&\color{blue}2&\color{blue}8\\ \color{red}1&\color{red}3&\color{red}13\\ \color{red}1&\color{red}3&\color{red}13\\ \color{green}1&\color{green}4&\color{green}14\\ \color{green}1&\color{green}4&\color{green}14\\ \color{yellow}1&\color{yellow}6&\color{yellow}20 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 6478: \(-1\), \(-1\), \(-1\), \(-2\), \(2\), \(\boldsymbol{0.71}\), \(\boldsymbol{-1.49}\), \(\boldsymbol{3.78}\)
Eigenvalues of 7028: \(-1\), \(1\), \(1\), \(-2\), \(-2\), \(\boldsymbol{0.71}\), \(\boldsymbol{-1.49}\), \(\boldsymbol{3.78}\)

CDC

Ryser Switch
\(G_{7028}=\operatorname{ryser}(G_{6478},14, 25)\)

Pair (6486, 7035)


Graph 6486

Graph 7035
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}11&\color{red}35\\ \color{red}1&\color{red}3&\color{red}11&\color{red}35\\ \color{blue}1&\color{blue}3&\color{blue}11&\color{blue}33\\ \color{red}1&\color{red}3&\color{red}11&\color{red}35\\ \color{red}1&\color{red}3&\color{red}11&\color{red}35\\ \color{green}1&\color{green}3&\color{green}9&\color{green}33\\ \color{green}1&\color{green}3&\color{green}9&\color{green}33\\ \color{yellow}1&\color{yellow}5&\color{yellow}15&\color{yellow}55 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 6486: \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{-0.34}\), \(\boldsymbol{0.66}\), \(\boldsymbol{-2.68}\), \(\boldsymbol{3.36}\)
Eigenvalues of 7035: \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{-0.34}\), \(\boldsymbol{0.66}\), \(\boldsymbol{-2.68}\), \(\boldsymbol{3.36}\)

CDC

Ryser Switch
\(G_{7035}=\operatorname{ryser}(G_{6486},14, 25)\)

Pair (6488, 7037)


Graph 6488

Graph 7037
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}14&\color{red}53\\ \color{red}1&\color{red}3&\color{red}14&\color{red}53\\ \color{blue}1&\color{blue}3&\color{blue}15&\color{blue}55\\ \color{red}1&\color{red}3&\color{red}14&\color{red}53\\ \color{red}1&\color{red}3&\color{red}14&\color{red}53\\ \color{green}1&\color{green}4&\color{green}16&\color{green}66\\ \color{green}1&\color{green}4&\color{green}16&\color{green}66\\ \color{yellow}1&\color{yellow}7&\color{yellow}23&\color{yellow}103 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 6488: \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{0.48}\), \(\boldsymbol{-1.53}\), \(\boldsymbol{-2}\), \(\boldsymbol{4.05}\)
Eigenvalues of 7037: \(-1\), \(1\), \(1\), \(\boldsymbol{0.48}\), \(\boldsymbol{-1.53}\), \(\boldsymbol{-2}\), \(\boldsymbol{-2}\), \(\boldsymbol{4.05}\)

CDC

Ryser Switch
\(G_{7037}=\operatorname{ryser}(G_{6488},14, 25)\)

Pair (10840, 10851)


Graph 10840

Graph 10851
\(\begin{pmatrix} \color{red}1&\color{red}3\\ \color{red}1&\color{red}3\\ \color{blue}1&\color{blue}4\\ \color{blue}1&\color{blue}4\\ \color{red}1&\color{red}3\\ \color{red}1&\color{red}3\\ \color{blue}1&\color{blue}4\\ \color{blue}1&\color{blue}4 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10840: \(1\), \(1\), \(1\), \(-2\), \(-2\), \(-2\), \(\boldsymbol{\frac{1}{2} \left(3-\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(3+\sqrt{17}\right)}\)
Eigenvalues of 10851: \(-1\), \(-1\), \(1\), \(-2\), \(-2\), \(2\), \(\boldsymbol{\frac{1}{2} \left(3-\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(3+\sqrt{17}\right)}\)

CDC

Ryser Switch
\(G_{10851}=\operatorname{ryser}(G_{10840},14, 36)\)

Pair (5358, 11716)

Note: This is the only pair which requires more than one Ryser switch.

Graph 5358

Graph 11716
\(\begin{pmatrix} \color{red}1\\ \color{red}1\\ \color{red}1\\ \color{red}1\\ \color{red}1\\ \color{red}1\\ \color{red}1\\ \color{red}1 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 5358: \(-1\), \(-1\), \(-1\), \(1\), \(1\), \(1\), \(-3\), \(\boldsymbol{3}\)
Eigenvalues of 11716: \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(\boldsymbol{3}\), \(\boldsymbol{3}\)

CDC

Ryser Switch
\(G_{11716}=\operatorname{ryser}(\operatorname{ryser}(G_{5358},16, 25),38,47)\)

Pair (10867, 11731)


Graph 10867

Graph 11731
\(\begin{pmatrix} \color{red}1&\color{red}3\\ \color{red}1&\color{red}3\\ \color{blue}1&\color{blue}4\\ \color{blue}1&\color{blue}4\\ \color{red}1&\color{red}3\\ \color{red}1&\color{red}3\\ \color{blue}1&\color{blue}4\\ \color{blue}1&\color{blue}4 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10867: \(0\), \(-1\), \(1\), \(-2\), \(\boldsymbol{\frac{1}{2} \left(3-\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(3+\sqrt{17}\right)}\), \(\frac{1}{2} \left(-1-\sqrt{17}\right)\), \(\frac{1}{2} \left(\sqrt{17}-1\right)\)
Eigenvalues of 11731: \(0\), \(-1\), \(-1\), \(-2\), \(\boldsymbol{\frac{1}{2} \left(3-\sqrt{17}\right)}\), \(\boldsymbol{\frac{1}{2} \left(3+\sqrt{17}\right)}\), \(\frac{1}{2} \left(1-\sqrt{17}\right)\), \(\frac{1}{2} \left(1+\sqrt{17}\right)\)

CDC

Ryser Switch
\(G_{11731}=\operatorname{ryser}(G_{10867},15, 26)\)

Pair (10841, 11742)


Graph 10841

Graph 11742
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}12\\ \color{red}1&\color{red}3&\color{red}12\\ \color{blue}1&\color{blue}4&\color{blue}16\\ \color{blue}1&\color{blue}4&\color{blue}16\\ \color{red}1&\color{red}3&\color{red}12\\ \color{red}1&\color{red}3&\color{red}12\\ \color{green}1&\color{green}5&\color{green}19\\ \color{green}1&\color{green}5&\color{green}19 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10841: \(1\), \(1\), \(-1-\sqrt{2}\), \(-2\), \(\boldsymbol{-0.32}\), \(\boldsymbol{-1.60}\), \(\boldsymbol{3.92}\), \(\sqrt{2}-1\)
Eigenvalues of 11742: \(-1\), \(-1\), \(-1-\sqrt{2}\), \(2\), \(\boldsymbol{-0.32}\), \(\boldsymbol{-1.60}\), \(\boldsymbol{3.92}\), \(\sqrt{2}-1\)

CDC

Ryser Switch
\(G_{11742}=\operatorname{ryser}(G_{10841},14, 36)\)

Pair (10838, 11747)


Graph 10838

Graph 11747
\(\begin{pmatrix} \color{red}1&\color{red}2&\color{red}8\\ \color{blue}1&\color{blue}4&\color{blue}14\\ \color{blue}1&\color{blue}4&\color{blue}14\\ \color{blue}1&\color{blue}4&\color{blue}14\\ \color{blue}1&\color{blue}4&\color{blue}14\\ \color{red}1&\color{red}2&\color{red}8\\ \color{green}1&\color{green}4&\color{green}16\\ \color{green}1&\color{green}4&\color{green}16 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10838: \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{0}\), \(\boldsymbol{0}\), \(\boldsymbol{\frac{1}{2} \left(1-\sqrt{41}\right)}\), \(\boldsymbol{\frac{1}{2} \left(1+\sqrt{41}\right)}\)
Eigenvalues of 11747: \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{0}\), \(\boldsymbol{0}\), \(\boldsymbol{\frac{1}{2} \left(1-\sqrt{41}\right)}\), \(\boldsymbol{\frac{1}{2} \left(1+\sqrt{41}\right)}\)

CDC

Ryser Switch
\(G_{11747}=\operatorname{ryser}(G_{10838},14, 36)\)

Pair (10839, 11748)


Graph 10839

Graph 11748
\(\begin{pmatrix} \color{red}1&\color{red}2&\color{red}8\\ \color{blue}1&\color{blue}4&\color{blue}16\\ \color{blue}1&\color{blue}4&\color{blue}16\\ \color{blue}1&\color{blue}4&\color{blue}16\\ \color{blue}1&\color{blue}4&\color{blue}16\\ \color{red}1&\color{red}2&\color{red}8\\ \color{green}1&\color{green}5&\color{green}21\\ \color{green}1&\color{green}5&\color{green}21 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10839: \(-1\), \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{0.21}\), \(\boldsymbol{-2.30}\), \(\boldsymbol{4.08}\)
Eigenvalues of 11748: \(-1\), \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{0.21}\), \(\boldsymbol{-2.30}\), \(\boldsymbol{4.08}\)

CDC

Ryser Switch
\(G_{11748}=\operatorname{ryser}(G_{10839},14, 36)\)

Pair (10849, 11751)


Graph 10849

Graph 11751
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}14&\color{red}56\\ \color{blue}1&\color{blue}4&\color{blue}17&\color{blue}69\\ \color{blue}1&\color{blue}4&\color{blue}17&\color{blue}69\\ \color{blue}1&\color{blue}4&\color{blue}17&\color{blue}69\\ \color{blue}1&\color{blue}4&\color{blue}17&\color{blue}69\\ \color{red}1&\color{red}3&\color{red}14&\color{red}56\\ \color{green}1&\color{green}4&\color{green}16&\color{green}68\\ \color{yellow}1&\color{yellow}6&\color{yellow}22&\color{yellow}96 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10849: \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{0.60}\), \(\boldsymbol{-1.35}\), \(\boldsymbol{-2.39}\), \(\boldsymbol{4.14}\)
Eigenvalues of 11751: \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{0.60}\), \(\boldsymbol{-1.35}\), \(\boldsymbol{-2.39}\), \(\boldsymbol{4.14}\)

CDC

Ryser Switch
\(G_{11751}=\operatorname{ryser}(G_{10849},14, 36)\)

Pair (10850, 11752)


Graph 10850

Graph 11752
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}15\\ \color{blue}1&\color{blue}4&\color{blue}19\\ \color{blue}1&\color{blue}4&\color{blue}19\\ \color{blue}1&\color{blue}4&\color{blue}19\\ \color{blue}1&\color{blue}4&\color{blue}19\\ \color{red}1&\color{red}3&\color{red}15\\ \color{green}1&\color{green}5&\color{green}23\\ \color{yellow}1&\color{yellow}7&\color{yellow}27 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10850: \(-1\), \(1\), \(1\), \(-2\), \(-2\), \(\boldsymbol{0.14}\), \(\boldsymbol{-1.64}\), \(\boldsymbol{4.50}\)
Eigenvalues of 11752: \(-1\), \(-1\), \(-1\), \(-2\), \(2\), \(\boldsymbol{0.14}\), \(\boldsymbol{-1.64}\), \(\boldsymbol{4.50}\)

CDC

Ryser Switch
\(G_{11752}=\operatorname{ryser}(G_{10850},14, 36)\)

Pair (10887, 11755)


Graph 10887

Graph 11755
\(\begin{pmatrix} \color{red}1&\color{red}3&\color{red}13\\ \color{blue}1&\color{blue}4&\color{blue}17\\ \color{blue}1&\color{blue}4&\color{blue}17\\ \color{blue}1&\color{blue}4&\color{blue}17\\ \color{blue}1&\color{blue}4&\color{blue}17\\ \color{red}1&\color{red}3&\color{red}13\\ \color{green}1&\color{green}5&\color{green}19\\ \color{green}1&\color{green}5&\color{green}19 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10887: \(-0.31\), \(-1\), \(1\), \(1.48\), \(-2.17\), \(\boldsymbol{-0.72}\), \(\boldsymbol{-2.38}\), \(\boldsymbol{4.10}\)
Eigenvalues of 11755: \(0.31\), \(-1\), \(-1\), \(-1.48\), \(2.17\), \(\boldsymbol{-0.72}\), \(\boldsymbol{-2.38}\), \(\boldsymbol{4.10}\)

CDC

Ryser Switch
\(G_{11755}=\operatorname{ryser}(G_{10887},14, 36)\)

Pair (10897, 11761)


Graph 10897

Graph 11761
\(\begin{pmatrix} \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{blue}1&\color{blue}6\\ \color{blue}1&\color{blue}6 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10897: \(0\), \(-1\), \(-1\), \(1\), \(1\), \(-2\), \(\boldsymbol{1-\sqrt{13}}\), \(\boldsymbol{1+\sqrt{13}}\)
Eigenvalues of 11761: \(0\), \(-1\), \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{1-\sqrt{13}}\), \(\boldsymbol{1+\sqrt{13}}\)

CDC

Ryser Switch
\(G_{11761}=\operatorname{ryser}(G_{10897},14, 36)\)

Pair (10898, 11762)


Graph 10898

Graph 11762
\(\begin{pmatrix} \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{red}1&\color{red}4\\ \color{blue}1&\color{blue}7\\ \color{blue}1&\color{blue}7 \end{pmatrix}\)

Walk Matrix

Eigenvalues of 10898: \(-1\), \(-1\), \(-1\), \(1\), \(1\), \(\boldsymbol{-2}\), \(\boldsymbol{-2}\), \(\boldsymbol{5}\)
Eigenvalues of 11762: \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(2\), \(\boldsymbol{-2}\), \(\boldsymbol{5}\)

CDC

Ryser Switch
\(G_{11762}=\operatorname{ryser}(G_{10898},14, 36)\)