最小Q-特征值第二小的给定悬挂点数的非二部单圈图

图的最小Q-特征值常被用来衡量一个图的非二部程度,受到研究者的广泛关注。在路Pn-k-2的一端接出一个圈C3,另一端接出k个悬挂边,所得的n阶图记为Ukn(3)。范益政等人最近证明Ukn(3)是最小Q-特征值达到最小的图。在他们的基础上,证明C13(n-k-1)是最小Q-特征值达到第二小的图,其中C13(n-k-1)是将Ukn(3)的一条悬挂边移至与悬挂邻点相邻的非悬挂点上所得的图。

The Non-bipartite Unicyclic Graph with Fixed Number of Pendent Vertices Whose Least Q-eigenvalue Attains the Second Smallest

The least Q- eigenvalue of a graph is often used to measure non-bipartiteness of the graph, which is widely concerned by researchers. Let U^k_n(3)be the graph of order n which is obtained by attaching a cycle C₃ to an end vertex of a path P_n-k-2 and attaching k pendant edges to the other end vertex of the path P_n-k-2. Recently, Yi-Zheng Fan et al. proved that U^k_n(3)was the unique graph whose least Q- eigenvalue attained the minimum among all graphs of order n with k pendant vertices. In this paper, we proved that C1₃(n-k-1) is the unique graph whose least Q- eigenvalue attains the second smallest value, where C1₃(n-k-1) is the graph obtained from U^k_n(3)by relocating a pendant edge to the non-pendant vertex which is adjacent the pendant neighbor.