Graphs with zero as a main eigenvalue of the signless Laplacian

Abstract

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 $\ell \ge 3$ $Q$-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 $k\ge 0$, 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.

arXiv comment12 pages, 4 figures. SageMath verification code and sample outputs are available at https://doi.org/10.5281/zenodo.21432139