Simple inexpensive vertex and edge invariants distinguishing dataset strongly regular graphs

Abstract

While standard Weisfeiler-Leman vertex labels are not able to distinguish even vertices of regular graphs, there is proposed and tested family of inexpensive polynomial time vertex and edge invariants, distinguishing much more difficult SRGs (strongly regular graphs), also often their vertices. Among 43717 SRGs from dataset by Edward Spence, proposed vertex invariants alone were able to distinguish all but 4 pairs of graphs, which were easily distinguished by further application of proposed edge invariants. Specifically, proposed vertex invariants are traces or sorted diagonals of $(A|_{N_a})^p$ adjacency matrix $A$ restricted to $N_a$ neighborhood of vertex $a$, already for $p=3$ distinguishing all SRGs from 6 out of 13 sets in this dataset, 8 if adding $p=4$. Proposed edge invariants are analogously traces or diagonals of powers of $\bar{A}_{ab,cd}=A_{ab} A_{ac} A_{bd}$, nonzero for $(a,b)$ being edges. As SRGs are considered the most difficult cases for graph isomorphism problem, such algebraic-combinatorial invariants bring hope that this problem is polynomial time.