关于双圈图的拟拉普拉斯谱半径的一个猜想的证明

Proof to a Conjecture on the Signless Laplacian Spectral Radius of Bicyclic Graphs

令q(G)表示图G的拟拉普拉斯谱半径.何春阳和郭曙光(2014)研究了不含三圈的n阶双圈图中拟拉普拉斯谱半径的排序问题,他们猜想"若n≥7,则q(G_(10))<q(G_9)",其中图G_9和G_(10)如图1所示.若该猜想成立,则其最终可以确定不含三圈的n≥12阶双圈图中排在前12位的拟拉普拉斯谱半径,该文证明了该猜想.

Let q （G） be the signless Laplacian spectral radius of G. He Chunyang and Guo Shuguang （2014） investigated the ordering problem of signless Laplacian spectral radii of bicyclic graphs with n verti- ces, which contains no triangles, they put forward the conjecture that ＂if n ≥ 7, then q （Gl0） 〈 q （G9）＂, where G9 and G10 are shown as in Figure 1. If this conjecture holds,then they have determined the first 12 largest signless Laplacian spectral radii of bicyclic graphs with n≥12 vertices,which contains no triangles. In this paper,the conjecture is proved.