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Graph isomorphism identification based on link-assortment adjacency matrix

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Abstract

Exploring a general and effective method of isomorphism identification is an arduous task. For this aim, some new concepts such as binary link path and link-assortment adjacency matrix are introduced in this paper. On this basis, a new method is proposed to improve the operability of isomorphism identification and relieve the computational pressure. By generating new elements from structural features and connection relations among links or vertices in the graph, it transforms traditional high-ranking adjacency matrix into new low-ranking adjacency matrix. Some typical graphs including kinematic chains, topological graphs, and planetary gear trains are used to verify the effectiveness of the proposed method. As shown by a comparison with results in the cited references, the proposed method is available in isomorphism identification.

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Author information

Authors and Affiliations

  1. College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou, 325035, People’s Republic of China

    Luchuan Yu, Hongbin Wang & Shunqing Zhou

Authors
  1. Luchuan Yu
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  2. Hongbin Wang
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  3. Shunqing Zhou
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Corresponding author

Correspondence to Luchuan Yu.

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Conflict of interest

Authors declare that they have no conflict of interest. All authors contributed to study conception and design. All authors read and approved the final manuscript.

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This article does not contain any studies with human participants performed by any of authors.

Appendices

Appendix A: Graphs and corresponding LAAMs for 9-link 2-DOF kinematic chains

Graphs and corresponding LAAMs for 9-link 2-DOF kinematic chains are listed in the Appendix A.

figure a

Chain1+5n

Chain2+5n

Chain3+5n

Chain4+5n

Chain5+5n

\(\left[ {\begin{array}{*{20}c} 3 & {} \\ {220} & {5\left( 3 \right)} \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 0 & 3 & {} \\ {30} & {21} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 4 & {} \\ {3211} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 1 & 3 & {} \\ {11} & {30} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 0 & 3 & {} \\ {11} & {31} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {21} & 3 & {} \\ 0 & 0 & {4\left( 3 \right)} \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ 3 & 0 & 3 & {} \\ 0 & 2 & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ 1 & 0 & 3 & {} \\ 0 & 3 & 1 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {11} & 3 & {} \\ 0 & 1 & {4\left( 3 \right)} \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ { - 1} & {30} & 3 & {} \\ {11} & { - 1} & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 2 & 3 & {} \\ {20} & {20} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 4 & {} \\ {3220} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 2 & 3 & {} \\ {11} & {20} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 2 & 3 & {} \\ {11} & {11} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 1 & 3 & {} \\ {20} & {21} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 0 & 3 & {} \\ {20} & {22} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ 2 & 0 & 3 & {} \\ 2 & 1 & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 1 & 3 & {} & {} \\ 1 & 0 & 3 & {} \\ 2 & 1 & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ 2 & 0 & 3 & {} \\ 2 & 0 & 1 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 1 & 3 & {} & {} \\ { - 1} & {20} & 3 & {} \\ {20} & { - 1} & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ { - 1} & {21} & 3 & {} \\ {20} & { - 1} & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 1 & 3 & {} \\ {21} & {11} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 0 & 3 & {} \\ {20} & {31} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 0 & 3 & {} \\ {11} & {22} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 1 & 3 & {} & {} \\ 2 & 0 & 3 & {} \\ 1 & 1 & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 1 & 3 & {} & {} \\ { - 1} & {20} & 3 & {} \\ {11} & { - 1} & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ { - 1} & {21} & 3 & {} \\ {11} & { - 1} & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 1 & 3 & {} & {} \\ {11} & { - 1} & 3 & {} \\ { - 1} & {11} & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 4 & {} \\ {2221} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 0 & 3 & {} \\ {21} & {21} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ 1 & 2 & 3 & {} \\ 2 & 0 & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ 1 & 2 & 3 & {} \\ 1 & 1 & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 2 & 3 & {} & {} \\ 1 & 1 & 3 & {} \\ 0 & 0 & 1 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 1 & 3 & {} \\ {30} & {20} & 4 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 1 & 3 & {} & {} \\ 1 & 1 & 3 & {} \\ 1 & 0 & 1 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {20} & 3 & {} \\ 0 & 1 & {4\left( 3 \right)} \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {30} & 3 & {} & {} \\ { - 1} & 0 & 3 & {} \\ 0 & { - 1} & {20} & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ 1 & 3 & 3 & {} \\ 0 & 1 & 0 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 0 & 3 & {} & {} \\ 3 & 0 & 3 & {} \\ 1 & 0 & 1 & 3 \\ \end{array} } \right]\)

\(\left[ {\begin{array}{*{20}c} 3 & {} \\ {211} & {5\left( 3 \right)} \\ \end{array} } \right]\)

  1. Note: n is an integer ranging from 0 to 7.

Appendix B: Graphs and corresponding LAAMs for 6-link 1-DOF planetary gear trains

Graphs and corresponding LAAMs for 6-link 1-DOF planetary gear trains are listed in the Appendix B.

figure b
figure c

G1+3n

G2+3n

G3+3n

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ {1_{2} } & { - 1} & {1_{2} } & {3^{2} } & {} \\ {2_{2}^{21} } & {1_{2} } & {2_{2}^{21} } & {2_{212} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} & {} \\ { - 1} & {2^{21} } & 3 & {} & {} & {} \\ {2^{2} } & { - 1} & { - 1} & 3 & {} & {} \\ { - 1} & {1_{2} } & {1_{2} } & {1_{2} } & {3^{2} } & {} \\ {1_{2} } & { - 1} & {1_{2} } & {2_{2}^{21} } & { - 1} & {3^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & 1 & 3 & {} & {} \\ {2^{2} } & { - 1} & 1 & 3 & {} \\ {1_{2} } & {2_{2}^{21} } & {1_{2} } & {2_{2}^{21} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} & {} \\ 1 & { - 1} & {2^{2} } & 3 & {} & {} \\ {1_{2} } & {2_{2}^{21} } & {1_{2} } & { - 1} & {3^{2} } & {} \\ {1_{2} } & {1_{2} } & { - 1} & {2_{2}^{21} } & { - 1} & {3^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2_{2}^{2} } & {3^{2} } & {} & {} \\ { - 1} & {1_{2} } & {2_{212}^{12} } & {3^{2} } & {} \\ {2^{21} 1} & { - 1} & {1_{2} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} & {} \\ 1 & 3 & {} & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & { - 1} & 3 & {} & {} \\ {1_{2} } & { - 1} & {2_{2}^{21} } & {1_{2} } & {3^{2} } & {} \\ {1_{2} } & { - 1} & {1_{2} } & {2_{2}^{21} } & { - 1} & {3^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ {1_{2} } & { - 1} & {2_{2}^{21} } & {3^{2} } & {} \\ {2^{21} } & 1 & 1 & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} & {} \\ 1 & 3 & {} & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & { - 1} & 3 & {} & {} \\ {1_{2} } & { - 1} & {2_{2}^{21} } & {2_{2}^{21} } & {3^{2} } & {} \\ {1_{2} } & { - 1} & {1_{2} } & {1_{2} } & { - 1} & {3^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ {1_{2} } & { - 1} & {1_{2} } & {3^{2} } & {} \\ {2^{21} } & 1 & {2^{21} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ {2_{2}^{21} } & { - 1} & {2_{2}^{21} } & {3^{2} } & {} \\ 1 & 1 & 1 & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2^{2} } & 3 & {} & {} \\ { - 1} & {2^{2} } & 3 & {} \\ {2^{21} 1} & 1 & {2^{21} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ 1 & { - 1} & {2^{2} } & 3 & {} \\ {1_{2} } & {2_{2} } & {1_{2} } & {2_{2}^{21} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ {2^{2} } & {2^{2} } & 3 & {} & {} \\ { - 1} & {2_{2} } & {1_{2} } & {3^{2} } & {} \\ {2^{21} 1} & {2^{2} } & { - 1} & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & 1 & 3 & {} & {} \\ 1 & { - 1} & {2^{2} } & 3 & {} \\ {1_{2} } & {2_{2}^{21} } & {1_{2} } & {2_{2}^{21} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & 1 & 3 & {} & {} \\ { - 1} & {1_{2} } & {2_{2}^{21} } & {3^{2} } & {} \\ {2^{21} 1} & { - 1} & {2^{2} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ { - 1} & 3 & {} & {} \\ {2^{2} } & {2^{21} 1} & 4 & {} \\ {2^{21} 1} & {2^{2} } & 1 & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ {2^{2} } & { - 1} & {2^{2} } & 3 & {} \\ {1_{2} } & {2_{2} } & {1_{2} } & {2_{2} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2_{2}^{21} } & {3^{2} } & {} & {} \\ 1 & {2_{2} } & 4 & {} \\ {2^{2} } & {1_{2} } & {2^{21} 2^{21} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2^{12} } & 3 & {} & {} \\ {2^{2} } & 1 & 4 & {} \\ {1_{2} } & {1_{2} } & {2_{2}^{21} 2_{2}^{21} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ {1_{2} } & {2_{2}^{21} } & {3^{2} } & {} & {} \\ {1_{2} } & {1_{2} } & { - 1} & {3^{2} } & {} \\ {2^{12} } & {2^{2} } & {1_{2} } & {2_{2}^{12} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ {1_{2} } & {2_{2}^{21} } & {3^{2} } & {} & {} \\ {1_{2} } & {1_{2} } & { - 1} & {3^{2} } & {} \\ 1 & {2^{2} } & {2_{2}^{12} } & {2_{2}^{12} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {1_{2} } & {3^{2} } & {} & {} \\ {2^{2} } & {2_{2}^{21} } & 4 & {} \\ {2^{21} } & {1_{2} } & {2^{21} 1} & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ 1 & 3 & {} & {} \\ 1 & {2^{2} } & 4 & {} \\ {1_{2} } & {2_{2}^{21} } & {2_{2}^{21} 2_{2}^{21} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ 1 & 3 & {} & {} & {} \\ { - 1} & {1_{2} } & {3^{2} } & {} & {} \\ {2_{2}^{21} } & {1_{2} } & { - 1} & {3^{2} } & {} \\ {2^{2} } & { - 1} & {2_{2}^{12} 2_{2}^{12} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ { - 1} & {3^{2} } & {} & {} \\ {2^{21} 1} & {1_{2} } & 4 & {} \\ {2^{2} } & {2_{2}^{12} 2_{2}^{12} } & 1 & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2_{2}^{21} } & {3^{2} } & {} & {} \\ 1 & {1_{2} } & 4 & {} \\ {2^{2} } & {2_{2}^{12} } & {2^{21} 1} & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2_{2}^{21} } & {3^{2} } & {} & {} \\ 1 & {1_{2} } & 4 & {} \\ {2^{2} } & {1_{2} } & {2^{12} 2^{12} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {2^{2} } & 4 & {} \\ {2^{21} 1} & {2^{21} 2^{21} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2^{21} } & 3 & {} & {} \\ 1 & {2^{2} } & 4 & {} \\ {1_{2} } & {1_{2} } & {2_{2}^{21} 2_{2}^{21} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {1_{2} } & {3^{2} } & {} & {} \\ {2^{2} } & {2_{2}^{12} } & 4 & {} \\ {2^{12} } & {1_{2} } & {2^{21} 1} & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ { - 1} & {3^{2} } & {} & {} \\ {2^{2} } & {2_{2}^{12} 2_{2}^{12} } & 4 & {} \\ {2^{12} 1} & {1_{2} } & 1 & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {2^{2} } & 4 & {} \\ {2^{12} 1} & {2^{21} 2^{21} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} & {} \\ { - 1} & {2^{21} } & 3 & {} & {} & {} \\ {2^{2} } & { - 1} & { - 1} & 3 & {} & {} \\ { - 1} & {1_{2} } & {1_{2} } & {1_{2} } & {3^{2} } & {} \\ {1_{2} } & { - 1} & {1_{2} } & {2_{2}^{21} } & { - 1} & {3^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ {1_{2} } & { - 1} & {1_{2} } & {3^{2} } & {} \\ {2^{12} } & 1 & {2^{21} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ {2^{2} } & {2^{2} } & 3 & {} & {} \\ {1_{2} } & {2_{2}^{21} } & { - 1} & {3^{2} } & {} \\ {2^{12} } & 1 & 1 & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ {2^{2} } & {2^{2} } & 3 & {} & {} \\ { - 1} & {2_{2}^{21} } & {1_{2} } & {3^{2} } & {} \\ {2^{12} 1} & 1 & { - 1} & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2^{2} } & 3 & {} & {} \\ { - 1} & {2^{2} } & 3 & {} \\ {2^{12} 1} & 1 & {2^{21} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2^{12} } & 3 & {} & {} \\ {2^{2} } & {2^{2} } & 4 & {} \\ {1_{2} } & {1_{2} } & {2_{2}^{21} 2_{2} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ { - 1} & {2^{12} } & 3 & {} & {} \\ {1_{2} } & {1_{2} } & {1_{2} } & {3^{2} } & {} \\ {2^{12} 1} & {2^{2} } & {2^{2} } & { - 1} & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} & {} \\ { - 1} & 1 & { - 1} & 3 & {} & {} \\ {1_{2} } & { - 1} & {2_{2}^{21} } & {2_{2}^{21} } & {3^{2} } & {} \\ {1_{2} } & { - 1} & {1_{2} } & {1_{2} } & { - 1} & {3^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & 1 & 3 & {} & {} \\ {1_{2} } & { - 1} & {1_{2} } & {3^{2} } & {} \\ {2^{21} } & {2^{2} } & {2^{21} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} & {} \\ {2^{2} } & { - 1} & {2^{12} } & 3 & {} & {} \\ 1 & 1 & { - 1} & { - 1} & 3 & {} \\ { - 1} & { - 1} & {1_{2} } & {1_{2} } & {1_{2} } & {3^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} & {} \\ 1 & 3 & {} & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {1_{2} } & { - 1} & {3^{2} } & {} & {} \\ {2^{2} } & { - 1} & {2^{2} } & {1_{2} } & 4 & {} \\ {1_{2} } & { - 1} & {1_{2} } & {2_{2} } & {2_{2}^{21} } & {4^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ {2^{2} } & 1 & 3 & {} & {} \\ {1_{2} } & {1_{2} } & { - 1} & {3^{2} } & {} \\ {2^{12} } & {2^{12} } & {2^{2} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {1_{2} } & {3^{2} } & {} & {} \\ {1_{2} } & { - 1} & {2_{212} } & {3^{2} } & {} \\ {2^{21} } & {2^{2} } & {2_{2}^{12} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ 1 & {2^{2} } & 3 & {} & {} \\ {1_{2} } & {1_{2} } & { - 1} & {3^{2} } & {} \\ {2^{12} } & {2^{21} } & {2^{2} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ {1_{2} } & {2_{2}^{21} } & {3^{2} } & {} & {} \\ {1_{2} } & {1_{2} } & { - 1} & {3^{2} } & {} \\ {2^{21} } & {2^{2} } & {1_{2} } & {2_{2}^{12} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {1_{2} } & {3^{2} } & {} & {} \\ {2^{2} } & {2_{2}^{21} } & 4 & {} \\ {2^{21} } & {1_{2} } & {2^{12} 1} & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2_{2}^{21} } & {3^{2} } & {} & {} \\ {2^{2} } & {1_{2} } & 4 & {} \\ 1 & {1_{2} } & {2^{21} 2^{12} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ {2_{2}^{21} } & { - 1} & {1_{2} } & {3^{2} } & {} \\ 1 & {2^{12} } & 1 & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {2^{2} } & 4 & {} \\ {2^{12} 1} & {2^{21} 2^{12} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ 1 & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ { - 1} & {1_{2} } & {2_{2} } & {3^{2} } & {} \\ {2^{12} 2^{12} } & { - 1} & {2^{2} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2_{2} } & {3^{2} } & {} & {} \\ {2^{2} } & {1_{2} } & 4 & {} \\ {2^{2} } & {1_{2} } & {2^{12} 2^{12} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & 1 & 3 & {} & {} \\ { - 1} & {1_{2} } & {2_{2}^{21} } & {3^{2} } & {} \\ {2^{12} 1} & { - 1} & {2^{2} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ { - 1} & 3 & {} & {} \\ {2^{21} 1} & {2^{2} } & 4 & {} \\ {2^{2} } & {2^{12} 1} & 1 & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ { - 1} & {1_{2} } & {2_{2} } & {3^{2} } & {} \\ {2^{12} 1} & { - 1} & {2^{2} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ {1_{2} } & {2_{2}^{21} } & {3^{2} } & {} & {} \\ {1_{2} } & { - 1} & { - 1} & {3^{2} } & {} \\ { - 1} & 1 & {1_{2} } & {2_{2}^{12} 2_{2}^{12} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ {1_{2} } & { - 1} & {1_{2} } & {4^{2} } & {} \\ 1 & 1 & 1 & {2_{2}^{12} 2_{2}^{12} } & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2_{2}^{21} } & {3^{2} } & {} & {} \\ 1 & {1_{2} } & 4 & {} \\ {2^{2} } & {1_{2} } & {2^{21} 2^{21} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {2^{2} } & 4 & {} \\ {2^{21} 1} & {2^{12} 2^{12} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ { - 1} & 3 & {} & {} & {} \\ { - 1} & {2^{12} } & 3 & {} & {} \\ {1_{2} } & {1_{2} } & {1_{2} } & {3^{2} } & {} \\ {2^{21} 1} & {2^{2} } & {2^{2} } & { - 1} & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ { - 1} & 3 & {} & {} \\ {2^{12} 1} & {2^{2} } & 4 & {} \\ {2^{2} } & {2^{12} 1} & 1 & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} {3^{2} } & {} & {} & {} \\ { - 1} & {3^{2} } & {} & {} \\ {1_{2} } & {2_{2}^{12} 2_{2}^{12} } & 4 & {} \\ {2_{2}^{12} 2_{2}^{2} } & {1_{2} } & 1 & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} \\ 1 & 4 & {} \\ {2^{12} 2^{12} } & {2^{21} 2^{21} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 5 & {} \\ {2^{21} 2^{21} 2^{12} 2^{12} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {1_{2} } & {3^{2} } & {} & {} \\ {2^{12} } & {1_{2} } & 4 & {} \\ {2^{2} } & {2_{2}^{12} } & {2^{12} 1} & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {2^{2} } & 4 & {} \\ {2^{12} 1} & {2^{21} 2^{12} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & {2^{2} } & 3 & {} & {} \\ {1_{2} } & { - 1} & {1_{2} } & {3^{2} } & {} \\ {2^{12} } & 1 & {2^{12} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2^{2} } & 3 & {} & {} \\ { - 1} & {2^{2} } & 4 & {} \\ {2^{12} 1} & 1 & {2^{12} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} & {} \\ {2^{2} } & 3 & {} & {} & {} \\ { - 1} & 1 & 3 & {} & {} \\ {1_{2} } & { - 1} & {1_{2} } & {3^{2} } & {} \\ {2^{12} } & {2^{2} } & {2^{21} } & {1_{2} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2_{2} } & {3^{2} } & {} & {} \\ {2^{2} } & {1_{2} } & 4 & {} \\ {2^{2} } & {1_{2} } & {2^{21} 2^{12} } & 4 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 5 & {} \\ {2_{2}^{21} 2_{2}^{21} 2_{2}^{21} 2_{2}^{21} 2_{2} } & {5^{2} } \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {1_{2} } & {4^{2} } & {} \\ {2^{12} 1} & {2_{2}^{12} 2_{2}^{12} 2_{2}^{12} } & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} {3^{2} } & {} & {} \\ {2_{2} } & 4 & {} \\ {2_{2}^{12} 2_{2}^{12} } & {2^{12} 2^{12} } & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} {3^{2} } & {} & {} \\ {1_{2} } & 4 & {} \\ {2_{2}^{12} 2_{2}^{12} } & {2^{12} 2^{12} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 5 & {} \\ {2^{12} 2^{12} 2^{12} 2^{12} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} \\ {1_{2} } & {4^{2} } & {} \\ {2^{21} 1} & {2_{2}^{12} 2_{2}^{12} 2_{2}^{12} } & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} {3^{2} } & {} & {} \\ {1_{2} } & 4 & {} \\ {2_{2}^{12} 2_{2}^{12} } & {2^{21} 2^{12} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 5 & {} \\ {2^{21} 2^{12} 2^{12} 2^{12} 1} & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2^{2} } & 3 & {} & {} \\ { - 1} & {1_{2} } & {3^{2} } & {} \\ {2^{12} 1} & 1 & {2_{2}^{12} 2_{2}^{12} } & 5 \\ \end{array} } \right]\)

\(L_{l} = \left[ {\begin{array}{*{20}c} 3 & {} & {} & {} \\ {2^{2} } & 3 & {} & {} \\ { - 1} & {1_{2} } & {3^{2} } & {} \\ {2^{12} 1} & 1 & {2_{2}^{12} 2_{2}^{12} } & 5 \\ \end{array} } \right]\)

  1. Note: n is an integer ranging from 0 to 26.

Appendix C: Graphs and corresponding LAAMs for 10-link co-spectral chains

Graphs and corresponding LAAMs for 10-link co-spectral chains are listed in the Appendix C.

figure d
$$ {\text{LAAM}}_{1} = \left[ {\begin{array}{*{20}l} {3^{2} } \hfill & {} \hfill & {} \hfill & {} \hfill \\ {1_{2} } \hfill & 3 \hfill & {} \hfill & {} \hfill \\ {20} \hfill & 0 \hfill & 4 \hfill & {} \hfill \\ {1_{22} } \hfill & {2_{2} } \hfill & {1_{2} } \hfill & { - 3^{2} } \hfill \\ \end{array} } \right]\;\;{\text{LAAM}}_{2} = \left[ {\begin{array}{*{20}l} 3 \hfill & {} \hfill & {} \hfill & {} \hfill \\ {1_{2} } \hfill & {3^{2} } \hfill & {} \hfill & {} \hfill \\ 0 \hfill & {20} \hfill & 4 \hfill & {} \hfill \\ {2_{2} } \hfill & {1_{22} } \hfill & {1_{2} } \hfill & { - 3^{2} } \hfill \\ \end{array} } \right]\;\;{\text{LAAM}}_{3} = \left[ {\begin{array}{*{20}l} 3 \hfill & {} \hfill & {} \hfill & {} \hfill \\ {1_{2} } \hfill & 3 \hfill & {} \hfill & {} \hfill \\ {20} \hfill & 0 \hfill & 4 \hfill & {} \hfill \\ {1_{22} } \hfill & {2_{2} } \hfill & {2_{2} } \hfill & { - 3^{2} } \hfill \\ \end{array} } \right] $$

Appendix D: Graphs and corresponding LAAMs for crossed 10-link kinematic chains

Graphs and corresponding LAAMs for crossed 10-link are listed in the Appendix D.

figure e
$$ {\text{LAAM}}_{1} = \left[ {\begin{array}{*{20}l} 3 \hfill & {} \hfill & {} \hfill & {} \hfill \\ {21} \hfill & 3 \hfill & {} \hfill & {} \hfill \\ { - 1} \hfill & 0 \hfill & 3 \hfill & {} \hfill \\ 0 \hfill & { - 1} \hfill & {21} \hfill & 3 \hfill \\ \end{array} } \right]\quad {\text{LAAM}}_{2} = \left[ {\begin{array}{*{20}l} 3 \hfill & {} \hfill & {} \hfill & {} \hfill \\ 2 \hfill & 3 \hfill & {} \hfill & {} \hfill \\ 1 \hfill & 0 \hfill & 3 \hfill & {} \hfill \\ 0 \hfill & 1 \hfill & 2 \hfill & 3 \hfill \\ \end{array} } \right] $$

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Yu, L., Wang, H. & Zhou, S. Graph isomorphism identification based on link-assortment adjacency matrix. Sādhanā 47, 151 (2022). https://doi.org/10.1007/s12046-022-01918-y

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