1. Mikhail (Mike, Misha) Kagan jointly with Misha Klin
Resistance-distance transform (RDT) in the context of Weisfeiler-Leman stabilization (WLS)
Mikhail (Mike, Misha) Kagan
jointly with Misha Klin
WL 2018

2. Outline Main idea of the approach RDT in a nutshell
Different approaches to RDT computations
Teaching example: WLS vs RDT
Some computations and observations
Conclusions
Further goals
References

3. Main idea of the approach
We start with a simple colored graph on n vertices
Using Kirchhoff’s rules (Kirchhoff, 1847), we compute so-called resistance distances (RD) between every pair of vertices
We stratify (now) complete graph Kn according to diverse values of RD’s
We obtain a coloring of Kn, which is considered as imitation of Weisfeiler-Leman stabilization (Weisfeiler, 1976)

4. RDT in a nutshell. Reminder

5. RDT in a nutshell. Examples
Rab = 2 a b
dab = 2
Rab = 1 a b
dab = 2
Rab = 2/3 a b
dab = 2

6. RDT in a nutshell. General definition of RD
Given electric network on n vertices with edge resistances Rij
Edge conductance
⟶ weighted adjacency matrix A
Complete colored graph Г with colors σij
E.g. simple graph: edge ⟷ σij = 1, non-edge ⟷ σij = 0
RD (i , j) = effective resistance b/w vertices i and j ,
provided ideal battery connected across i and j
Resistance Distance Transform
A ⟶ Ã = new weighted adjacency matrix with entrees
0 if i = j and 1/RD(i, j) otherwise

7. RDT in a nutshell. RD as a metric
Resistance distance satisfies (Klein & Randič, 1993)
R (a, b) ≥ 0
R (a, b) = 0, iff a=b
R (a, b) = R (b, a)
R (a, b) ≤ R (a, c) + R (c, b)
For a tree: RD = standard distance
The symmetry of RD will be of one the reasons for limitation of the approach

8. Different approaches to RD computations
Provided
Adjacency matrix: Aij = Aji
Degree matrix: Dij = diag (d1, d2, …), di = ΣjAij
Laplacian matrix: L = D - A
Resistance Distance
Method 1 (Klein & Randič, 1993)
where Гij = Moore-Penrose pseudo inverse of Laplacian matrix
Method 2 (Bapat, 1999; MK, 2015)

9. Different approaches to RD computations
Method 3 (Bapat & Gupta, 2010)
2-forest separating i and j = forest with n – 2 edges and two connected components, such that i and j belong to different components
Example 1 2 3 1 2 3 1 2 3 1 2 3
One tree
One 2-forest for 1 & 2
Two 2-forests for 1 & 3
RD (1,2) = RD (2,3) =1
RD (1,3) = 2

10. Teaching Example: WLS vs RDT
WLS procedure
Non-commutative variables:
a for each vertex
b for each edge
c for each non-edge
Replace distinct polynomials with new distinct variables
Repeat until stabilization: Mk+1 ≈ Mk . In this example k = 2.

11. Teaching Example: WLS vs RDT
RDT procedure
Need to compute:
RD (1,2) RD (1,3)
RD (2,4)
New Laplacian matrix (RDT of L)

12. Teaching Example: WLS vs RDT
Comparison of WLS and RDT results
L2 = symmetrized version of M2

13. Some computations and observations Distance regular graphs
RDT was performed for a number of connected DRG’s (Г). In each case, the number of distinct colors was equal to the diameter d of Г.
In addition, obtained ”patterns” of RD’s for algebraically isomorphic, but not isomorphic DGR’s, were also equivalent.
The computations were performed for :
Shrikhande and L2(4) on 16 vertices
15 and 10 Paulus graphs on 25 and 26 vertices
Chang graphs on 28 vertices
Dodecahedron (d=5) on 20 vertices

14. Some computations and observations Two Brouwer’s {0,2}-graphs
Two {0, 2}-graphs (in the sense of Andries Brouwer) on 20 vertices were considered:
The vertex-transitive N6.3 and
The N6.4 with Aut of order 24
In both cases, the result of symmetrization of the WL-closure coincided with RDT
For N6.3, RDT yields Jordan closure (in the sense of P. Cameron)

15. Some computations and observations
Co-spectral example: 4-Cube and Hoffman
For the 4-Cube (Q4) of diameter 4, standards results are obtained: RDT = WLS
For Hoffman graph, which is co-spectral to Q4, the WL-closure has rank 19, with 3 fibers. Its symmetrization has rank 11.
The first iteration of RDT yields 2 fibers with 6 colors.
The result of the second iteration of RDT coincides with the symmetrization of WL closure: 3 fibers and 11 colors

16. Some computations and observations Two Circulant graphs on 20 vertices
There are two non-isomorphic co-spectral circulants on 20 vertices, both generate the same coherent closure
We get the same “patterns” applying RDT, however the two graphs can be distinguished by their distinct sets of RD values

17. Conclusions RDT provides an intriguing approach to imitate WLS
Comparison with WLS:
For certain graphs (e.g. DRG’s) RDT leads to the same result (corollary of theorems by Norman Biggs [3])
RDT often stabilizes in one iteration, provided that minimal degree (valency) is at least 2
Limitations
In considered examples, RDT provides symmetrized version of WLS
Sometimes requires more than one iteration (Hoffman)

18. Future goals To perform more experiments and to interpret them
To prove rigorously that RDT(Г ) is a part of WL(Г )
To understand when RDT is a Jordan scheme
To generalize the results of Biggs to AS’s

19. References
[1]  R. Bapat, “Resistance distance in graphs,’’ Mathematics Student, Vol 68, Nos. 1-4 (1999)
[2]  R. Bapat and S. Gupta, “Resistance distance in wheels and fans,’’ Indian J. Pure Appl. Math, 41(1), 1-13 (2010)
[3] N. Biggs, “Potential Theory on Distance-Regular Graphs”, Combinatorics, Probability and Computing, 2(3), (1993)
[4]  M. Kagan, “On equivalent resistance of electrical circuits,’’ Am. J. Phys. 83, (2015)
[5] G. Kirchhoff, “Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strömegeführt wird,” Ann. Phys. Chem. 148, 497–508 (1847)
[6]  D. J. Klein and M. Randič, “Resistance distance,’’ J. Math. Chem., Vol 12, (1993)
[7] B. Weisfeiler (editor), “On construction and identification of graphs’’. Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin- New York, xiv+237 pp.

20. Thank you for your attention!
RDT = symmetrized version of WLS