Improved high-order adjacent vertex assignment sequence for similarity vertices and isomorphism identification of planar kinematic chains

 Abstract: Isomorphism identification is fundamental to synthesis and innovative design of kinematic chains (KCs). The identification can be performed accurately by using the similarity of KCs. However, there are very few researches on isomorphism identification based on the properties of similarity vertices. In this paper, an improved high-order adjacent vertex assignment (IHAVS) sequence method is proposed to seek out the similarity vertices and identify the isomorphism of the planar KCs. First, the specific definition of IHAVS is described. Through the calculation of the IHAVS, the adjacent point value sequence reflecting the uniqueness of the structural features is established. Based on the value sequence, all possible similarity vertices, corresponding relations and isomorphism discrimination can be realized. By checking the topological diagrams of KCs of different number of links, the correctness of the proposed method are verified. Finally, the method is used to find the similarity vertices of all the 9-link 2-DOF(degree of freedom) planar


Introduction
At present, the multi-component and multi-freedom mechanism is widely used in various mechanical equipment, involving engineering machinery, medical machinery, agricultural machinery, etc. The innovative design of mechanism structure is still the key step of product upgrading [1][2]. The correct solution of the similarity vertices [3] can not only reduce the number of isomorphism identification, but also provide a theoretical basis for the selection of rack, input and output components. In 1992, Yan [4] proposed a new mechanism design technology, called regenerative KCs method, which essentially represents the logical reasoning of the regeneration process of KCs. In logic reasoning, the similarity vertices should be analyzed to improve the efficiency of innovative design and reduce redundant design scheme. In the same year, Hwang [5] divided the similarity into symmetry similarity, transfer similarity, row similarity and irregular similarity, and proposed the relationship code method of weighted line graph to distinguish the similarity vertices of functional components. Although Hwang's method is simple, it has the situation of misjudgment.
In mechanism synthesis, isomorphism identification is fundamental. Many researches have been carried out and a large number of methods and theories have been developed. Kong et al. [6] first applied the artificial neural network technology to isomorphism identification, and established the neural network model of topological graph. Chang et al. [7] proposed to identify KCs isomorphism by comparing the eigenvalues and eigenvectors of the corresponding adjacency matrix of topological graphs for the first time; Cubillo and Wan [8] found that the original theory had errors, proposed the necessary and sufficient conditions for eigenvalues and eigenvectors of adjacent matrices of isomorphic KCs to modified the methods; Later, Sunkari and Schmidt [9] found that this method has certain limitations. This method fails when the number of components in the KCs is greater than 14. Xiao et al. [10] creatively combined ant colony algorithm and artificial immune algorithm to identify the isomorphism of KCs. Ding and Huang [11][12][13][14] standardized topological graph according to certain rules, used the normal adjacency matrix to identify the KCs isomorphism, and developed the algorithm of related isomorphism identification. Galán-Marín et al. [15] first used the multi value neural network method to identify the isomorphism of KCs. In the same year, Dargar et al. [16] proposed to identify isomorphism by comparing the first-order and second-order adjacent component values of the KCs, but Improved high-order adjacent vertex assignment sequence for similarity vertices and isomorphism identification of planar kinematic chains ·3· this method has the defect that it can't describe the uniqueness of each vertex in the topological graph. Yang et al. [17] proposed a method of detecting isomorphism based on the correlation matrix of KCs. By using this method, the sufficiency of isomorphism identification was obtained on the basis of vertex correspondence. Zeng et al. [18] proposed a partition matching algorithm to the identify isomorphism. Yang et al. [19] proposed a hybrid immune algorithm by combining immune algorithm, genetic algorithm and local search algorithm, and realized the recognition of isomorphism by using this algorithm. Shang et al. [20] proposed to use the method of optimized circuit simulation to identify the KCs isomorphism. In 2018, Sun et al. [21] proposed to use joint matrix to uniquely describe the structure of KCs, obtain the corresponding component and joint information from joint matrix, and use joint matrix, component and joint to identify isomorphism. In the same year, Rai and Punjabi [22] first used connection number and entropy to ignore tolerance gap to identify isomorphism. He et al. [23] used the dynamic IHAVS to find the similarity vertices and distinguish the isomorphism of the topological graph. However, when using its IHAVS to find the similar points, there are counter examples, such as Figure 3 and the topological diagram with the serial number of 9 in the appendix.
Although all the above synthesis methods can identify the isomorphism, they can't accurately find the similarity vertices that are beneficial for mechanism selection and isomorphism identification. In order to solve this problem, a method is proposed to find the similarity vertices and identify the isomorphism of the KCs based on the improved IHAVS sequence. And the correctness of this method is verified.
The rest of this paper is organized as follows: Section 2 introduces the basic knowledge of graph theory. In Section 3, the initial value rule and the definition of IHAVS of topological graph are described. Several examples are given to find the similarity vertices of KCs. In Section 4, based on the correct solution of the similarity vertices, the concrete steps of identifying the isomorphism of the KCs are given. Section 5, conclusion.
The degree of vertex i in topological graph is defined as the number of edges connected with vertex i; it can also be determined by the sum of i-th column (row) of adjacency matrix A. As shown in Figure 1, the topological diagram of the 9-link 2-DOF KC is expressed as (b), and the corresponding adjacency matrix is A. Vertices 2, 4, 6, 7 and 9 are two-dimensional points, while vertices 1, 3, 5 and 8 are three-dimensional points.

Definition of similarity vertices
We consider a graph as labeled when its vertices are labeled by the integers 1, 2, …, n. In this regard, a labeled graph is mapped into another labeled graph when the n integers are permuted. For some permutations, a labeled graph may map into itself. A set of those permutations, which map the graph into itself, form a group that is a group of automorphisms. It is also said to be a vertex-induced group. Two vertices of a graph contained in the same permutation cycle of a vertex-induced group of automorphisms are deemed similar.
As shown in Figure 1 (b), based on the component similarity classification proposed in reference [5], vertices 4 and 9, 5 and 8, 6 and 7 are symmetrically similar, the possible number of automorphism groups is 2! × 2! × 2 = 8, and listed as follows:  p and 8 p can transform the adjacency matrix into itself. The chain automorphism group is 18 { , } pp . Vertices 4 and 9, 5 and 8, 6 and 7 are in the automorphism cycle 8 p and are thus similar.

Isomorphism of KCs
If one-to-one mapping f exists for the two graphs

Initial value and IHAVS
For any given topological graph, the number of 3j of Fibonacci sequence [24][25] is assigned according to the size of each vertex degree, where j is a positive integer. According to the vertex degrees, the values of j are 1, 5, 21, 89.... The vertices with equal degree are assigned the same value. This assignment method is defined as the initial value, and the initial value sequence is recorded as 0 S . As shown in Figure 1 (b), since vertices 2, 4, 6, 7 and 9 are two-dimensional points, the initial value of vertices 2, 4, 6, 7 and 9 is 1. Vertices 1, 3, 5 and 8 are three-dimensional points, the initial value of vertices 1, 3, 5 and 8 is 5, that is, the initial value sequence is 0 Note that the r-th order adjacent vertex value (AVV) of vertex i in topological graph G is r i S : 11 ,, 11 1 10 where n is the total number of vertices in the topology.
, ij a is the element corresponding to row i and column j of adjacency matrix A.
It can be seen from the analysis that the larger r is, the more fully r-order AVV characterizes the vertex of topological graph. And the larger r is, the more complex its calculation is. So it is of great significance to choose the right r. In this paper, r is taken as the integral part of nd + , where d changes dynamically according to the specific situation.
The weight of the weighting part is 1/10 instead of 1/n. The advantage of this value is that the decimal point of the AVV of order r is exactly equal to r, which can avoid rounding error.
The sequence of the r-order AVV of all vertices is The r-th order AVV of vertex i is determined by the initial value of vertex i, degree of vertex i, the degree of the vertex adjacent to vertex i and the (r-1)-th order AVV of the vertex adjacent to vertex i. Hence, the r-th order AVV can uniquely describe the characteristics of each vertex in the topological graph. That is, in the r-th order AVV sequence r S , if the r-th order AVV of the two vertices are the same, the two vertices are similar, otherwise they are not similar. The solution of similarity vertices is of great significance to the selection of frame, input and output components. The correct solution of similarity vertices can reduce the number of isomorphism identification and improve the overall efficiency of mechanism synthesis. Improved high-order adjacent vertex assignment sequence for similarity vertices and isomorphism identification of planar kinematic chains ·5· 3.2 Examples of finding similarity vertices Example 1: As shown in Figure 3, the corresponding topological diagram of the 11-link 2-DOF KCs (a) is shown in Figure (b). The process of finding the similarity vertices by using the IHAVS before and after correction is as follows: IHAVS before modification [23]: The value of r before correction is 4, the initial value of the prime number sequence according to the vertex degree of the topological graph is recorded as 0 S , and the fourth order AVV sequence is recorded as S4, the solution results before correction are shown in Table 1. Table 1 The initial value and fourth order AVV sequence before modification of Figure 3 Sequence , vertices 6 and 9 are similar. But in fact, there is no similarity vertices in this topological diagram. Considering whether the value of r is too small, which leads to the correctness of the conclusion, we have solved the similarity of the fifth order, eighth order and even eleventh order AVV series. The conclusion is still that 6 and 9 are similar, so there are problems in the formula before modification.
Improved IHAVS sequence: total vertices of topology n=11, 0 14 rn = +  , 1 25 rn = +  . The initial value of vertex degree in topological graph is S0. By using the initial value, the fourth and fifth order AVV sequence obtained from equation (3) is recorded as S4 and S5, the specific solution results after correction are shown in Table  2. Table 2 The modified initial value and r0, r1-th order AVV sequence of Figure 3 Sequence According to the fourth and fifth order AVV sequence S4 and S5, the adjacency values of each vertex are different, so there are no similar vertices in the topological graph. Therefore, in the innovative design of the mechanism, there are 11 different ways to choose the frame.
Example 2: As shown in Figure 4, there is a topology with 12 vertices and 15 edges. The solution process of the similarity vertices is as follows: topological graph is S0; By using the initial value, the fourth and fifth order AVV sequence obtained from equation (3) is recorded as S4 and S5, the specific solution results after correction are shown in Table 3: Table 3 The initial value and r0, r1-th order AVV sequence of Figure 4 Sequence . Vertices 1 and 6, 4 and 11, 5 and 10, 7 and 9 are similar to each other; Vertices 2, 3, 8 and 12 have different adjacency values with other points, so they have no similarity vertices. Therefore, in the innovative design of the mechanism, there are 8 different ways to choose the frame.
Example 3: As shown in Figure 5, there is a topology with 21 vertices and 29 edges. The solution process of the similarity vertices is as follows: The initial value of vertex degree in topological graph is S0. By using the initial value, the fifth and sixth order AVV sequence obtained from equation (3) is recorded as S5 and S6, the specific solution results after correction are shown in Table 4: Table 4 The initial value and r0, r1-th order AVV sequence of    Improved high-order adjacent vertex assignment sequence for similarity vertices and isomorphism identification of planar kinematic chains ·7· Therefore, in the innovative design of the mechanism, there are 13 different ways to choose the frame.
In the end of this paper, we use the IHAVS sequence to find the similarity vertices of all the 9-link 2-DOF topological diagrams. See the appendix for the specific solution results, which further verifies the correctness of this method.

4
Steps and examples of isomorphic For any given two topological graphs, if they are isomorphic, their corresponding vertex degrees must be the same, and the vertex degrees at the same distance from the two corresponding vertices must be the same. It can be seen from equation (3) that the r-th order AVV of vertex i is determined by the initial value of vertex i, degree of vertex i, the degree of the vertex adjacent to vertex i and the (r-1)-th order AVV of the vertex adjacent to vertex i. Therefore, the corresponding relationship can be found by the r-order AVV sequence r a S and r b S . The adjacency matrix of topological graph G(a) is transformed first by row and then by column by corresponding relationship. Then, the adjacency matrix after transformation is compared with the adjacency matrix of topological graph G(b). If they are the same, they are isomorphic. Otherwise, they are non-isomorphic. The specific steps of isomorphism identification are as follows: Step1: For any given two topological graphs G(a) and G(b), the initial values of each vertex degree are recorded as 0 are calculated by equation (3) to distinguish the similarity vertices between G(a) and G(b). If the number of groups of two topological graphs is different or the number of groups is the same but the number of similar points of corresponding groups is different, the two topological graphs are non-isomorphic. Otherwise, go to step 2. If there are no similarity vertices between the two topologies and the sequence r a S and r b S has no corresponding relationship, the two topologies are non-isomorphic. Otherwise, there is a set of corresponding relationship b a M , go to step 5.
Step2: The indefinite value of sequence r a S is assigned according to the following rules: Select the vertex with few similar points and the smallest label as i, change its initial value to the sum of the original initial value and the (p+1) Fibonacci number, where p is the number of indefinite assignment. For example, the vertex selected for the first initial value is j, its corresponding adjacent point value is recorded as t1, the new initial value of vertex j is recorded as g1, and its new r-order AVV sequence calculated by equation (3)  Step3: According to T=(t1,t2,…,tp) and G=( g1, g2,…,gp) to assign a fixed value to sequence r b S , the specific rules are as follows: Obtain the same vertex as the first element t1 in the replacement set T in the sequence r b S , and assign a new initial value to this vertex as the first element g1 in the replacement set G, the new r-order AVV sequence Step5: According to the corresponding relation, the adjacency matrix A corresponding to the topological graph G(a) is transformed first by row and then by column, one or more transformed adjacency matrix A2 is obtained.

·8·
Compared A2 with the adjacency matrix B corresponding to the topological graph G(b), if there is no corresponding, the two topological graphs are non-isomorphic and the program terminates; otherwise, the two topological graphs are isomorphic.

Examples of isomorphic identification
Example 1: As shown in Figure 6, there are 11-link 2-DOF topological diagrams G(a) and G(b) and its adjacency matrices are A and B respectively. The isomorphism identification process is as follows:  Table 5. It can be seen from the sequence 5 a S and 5 b S that there is no similarity between G(a) and G(b), but there is a corresponding relationship between the sequence 5 a S and 5 b S . The corresponding relationship b a M = {1、8、7、6、 4、3、2、5、9、10、11}.
Step5: On the basis of b a M , the adjacency matrix A corresponding to the topological graph G(a) is transformed, first by row and then by column, into an adjacency matrix A2. Compared A2 with the adjacency matrix B corresponding to the topological graph G(b), they are the same, so the two topological graphs are isomorphic.
Example 2：As shown in Figure 7, G(a) and G(b) are topological diagrams with 30 vertices and 40 edges. Isomorphism identification process is as follows:  Table 6. S are equal to 142.6602420, assign values to these two positions respectively to obtain the seventh order AVV sequence 7 ,1, bq S ,where q=1 represents the adjacency value sequence obtained by changing the initial value of vertex 21 in topological graph G(b), q=2 represents the adjacency value sequence obtained by changing the initial value of vertex 25 in topological graph G(b), and the final sequence result is shown in Table 7.
Step5: On the basis of b a M , the adjacency matrix A corresponding to the topological graph G(a) is transformed first by row and then by column, adjacency matrix A2 is obtained. Compared A2 with the adjacency matrix B corresponding to the topological graph G(b),they are the same, so the two topological graphs are isomorphic. of the seventh order AVV sequence are shown in Table 8. Step4: Because none of the six final sequences can correspond to the sequence 5 ,2 a S , the two topologies are non-isomorphic.

Conclusions
In this paper, an improved IHAVS sequence method is proposed not only to find similarity vertices to improve the efficiency of isomorphism identification, but also to provide a theoretical basis for selecting the functional parts of the mechanism. The IHAVS sequence is employed to describe the uniqueness of each vertex in topological graph. Further, with the IHAVS sequence, the similarity vertices can be found accurately, and the reliability of isomorphic discrimination can be guaranteed by confirming the similarities. In order to prove the correctness of this method, the topological diagrams of 9, 11, 21 and 30 links KCs have been checked. Moreover, the similarity library of 9-link 2-DOF KCs is given for the first time. Our method improves the overall efficiency of mechanism synthesis.