A coherent algebra is a matrix algebra (over $\mathbb{C}$) closed under conjugate transpose and Schur (entrywise) product, and that contains the identity matrix $I$ and all ones matrix $J$. Given coherent algebras $\mathcal{A}$ and $\mathcal{B}$, a weak isomorphism from $\mathcal{A}$ to $\mathcal{B}$ is a bijective linear map $\varphi$ satisfying:

  1. $\varphi(MN) = \varphi(M)\varphi(N)$;
  2. $\varphi(M \circ N) = \varphi(M) \circ \varphi(N)$ (where $\circ$ denotes the Schur product);
  3. $\varphi(M^*) = \varphi(M)^*$ where $^*$ denotes conjugate transpose;
  4. $\varphi(J) = J$.

Given a graph $G$ with adjacency matrix $A$, the coherent algebra of $G$ is defined as the intersection of all coherent algebras containing $A$. In this paper they define two graphs $G$ and $H$ to be equivalent if there exists a weak isomorphism between their corresponding coherent algebras that also maps the adjacency matrix of $G$ to the adjacency matrix of $H$. In the paper they say that there exists a polynomial time algorithm for testing whether two graphs are equivalent and cite Weisfeiler and Lehman's paper.

I gather that the Weisfeiler-Lehman algorithm is closely related to coherent algebras, but have so far only been able to find references which describe the algorithm in a purely combinatorial manner, i.e. in terms of graphs not coherent algebras. The paper of Weisfeiler and Lehman linked above is not available to me and is anyways written in Russian. I am wondering if there is a direct connection with the definition of graph equivalence given above, and the Weisfeiler-Lehman algorithm for distinguishing graphs. More specifically, is it true that two graphs are equivalent if and only if they cannot be distinguished by ($k$-dimensional) Weisfeiler-Lehman (for some [specific] $k$)?

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    $\begingroup$ Maybe this paper/preprint could be relevant for you: Babel, Baumann, Ludecke, Tinhhofer: STABCOL -- Graph isomorphism testing based on the Weisfeiler-Leman algorithm. $\endgroup$ – Stefan Gyürki Sep 30 '15 at 9:49
  • $\begingroup$ Both the original paper and its translation are now available here. iti.zcu.cz/wl2018/wlpaper.html $\endgroup$ – Daniel Aug 24 '20 at 8:55

A standard reference is "On Construction and Identification of Graphs", ed. by: B.Weisfeiler, Springer Lect. Notes Math. Vol. 558 (1976) ISBN: 978-3-540-08051-0

There you will find details on Weisfeiler-Lehman algorithm (called stabilization there, IIRC). Although it's not an easy read.

A modern reference with some details on various "generalised" isomorphisms is here.

  • $\begingroup$ The link in the last sentence seems not to work anymore. Could you please update it or provide the title of the work you mention? $\endgroup$ – Daniel Aug 24 '20 at 8:51
  • $\begingroup$ I most probably meant a version of dl.acm.org/doi/10.1145/2371656.2371662 - not 100% sure though. $\endgroup$ – Dima Pasechnik Aug 24 '20 at 15:38

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