Characterization of singular simple graphs of fixed order

Characterize all simple graphs of order n whose adjacency matrix has rank strictly less than n, thereby resolving the longstanding singular-graph characterization problem.

Background

The paper places its study of row left rank in the broader research program concerning graph rank and singularity. It notes that Collatz and collaborators initiated the problem of characterizing, for a given order n, all simple graphs whose adjacency matrix is singular. This problem is presented as a foundational unresolved question motivating subsequent work on rank and nullity bounds under graph parameters such as matching number, pendant-vertex number, maximum degree, and girth.

References

Collatz et al. first proposed to characterize all graphs of order $n$ with $r(G)<n$. Until today, this problem is still unsolved.

— The row left rank of quaternion unit gain graphs in terms of pendant vertices  (2504.07784 - Lu et al., 10 Apr 2025) in Section 1, Introduction