以无符号拉普拉斯矩阵的零特征值为主特征值的图
Graphs with zero as a main eigenvalue of the signless Laplacian
AI总结：
研究刻画恰好有\(\ell\geq3\)个\(Q\)-主特征值且其一为零的图，给出\(\ell = 3\)时的情况，还构造了无穷多个圈秩为\(k\)且直径无界的图，为相关图的分类提供反例。
AI中文摘要：
无符号拉普拉斯矩阵\(Q(G)\)的一个特征值若其特征空间不与全一向量正交，则称其为\(Q\)-主特征值。我们刻画了恰好有\(\ell\geq3\)个\(Q\)-主特征值且其中一个为零的图。\(\ell = 3\)的情况归结为满足顶点加权度和恒等式的非半正则二分图。对于每个整数\(k\geq0\)，我们构造了无穷多个圈秩为\(k\)且直径无界的两两非同构的图，它们都恰好有三个包括零的\(Q\)-主特征值。这些族为Javarsineh和Fath - Tabar对树、单圈图和双圈图的分类提供了反例。
英文摘要：
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.
