Graphs with zero as a main eigenvalue of the signless Laplacian

An eigenvalue of the signless Laplacian $Q(G)$ is $Q$-main if its eigenspace is not orthogonal to the all-ones vector.

We characterize graphs with exactly $\ell\ge3$ $Q$-main eigenvalues, one of which is zero.

The case $\ell=3$ reduces to non-semiregular bipartite graphs satisfying a vertexwise signed degree-sum identity.

For each integer $k\ge0$, we construct infinitely many pairwise nonisomorphic graphs of cyclomatic number $k$ and unbounded diameter, all with exactly three $Q$-main eigenvalues including zero.

These families provide counterexamples to the stated classifications of trees, unicyclic graphs, and bicyclic graphs of Javarsineh and Fath-Tabar.