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arXiv · 2607.17805

Graphs with zero as a main eigenvalue of the signless Laplacian

Hangxi Cha·Haiying Shan
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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 $\ell\ge3$ $Q$-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.

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Hangxi Cha, Haiying Shan. 2026-07-20. Graphs with zero as a main eigenvalue of the signless Laplacian. https://arxiv.org/abs/2607.17805

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