The 
-dimensional
Weisfeiler-Leman algorithm is an iterative graph coloring
algorithm that assigns colors to 
-tuples of vertices of a finite
graph. For 
, the initial color of a tuple records the equality and
adjacency relations among its coordinates. At each subsequent round, the new color
records the old color together with the multiset, over
all vertices 
,
of the ordered 
-tuples of colors obtained by replacing each coordinate in
turn by 
.
The process stops when no color class splits (Immerman and Sengupta 2019). The one-dimensional
Weisfeiler-Leman algorithm is color refinement.
To compare graphs 
and 
, the algorithm is applied to their disjoint
union. It distinguishes the graphs if the multisets
of stable colors on the 
-tuples from 
and 
are different. Since the coloring is invariant under graph
isomorphism, graphs distinguished by the algorithm cannot be isomorphic. The
least 
for which the algorithm distinguishes 
from every nonisomorphic graph is the Weisfeiler-Leman
dimension of 
(Grohe 2017).
The algorithm is named for Boris Weisfeiler and Andrei Leman (1968), who introduced the original method while reducing a finite multigraph
to canonical form. Their procedure repeatedly refines
colors on ordered pairs of vertices by counting two-step
walks with specified color sequences. This pair refinement
is the classical two-dimensional Weisfeiler-Leman algorithm. The modern 
-dimensional algorithm extends it from ordered pairs to tuples
of 
vertices (Grohe 2017). At stabilization, the adjacency matrices of the color classes
span an associative algebra 
. This is the "algebra which appears therein"
in the paper's title. Its relation to the automorphism
group of 
was central to the authors' study.
The stable coloring produced by the two-dimensional algorithm determines a coherent configuration.
-Dimensional Weisfeiler-Leman Algorithm." 22 Jul 2019.