2026/07/20 by Hangxi Cha, Haiying Shan
#math.CO
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 ℓ≥3 Q-main eigenvalues, one of which is zero. The case ℓ=3 reduces to non-semiregular bipartite graphs satisfying a vertexwise signed degree-sum identity. For each integer k≥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.