图C_m^k(P_2,…,P_2,P_l)的最大特征值

The Largest Eigenvalue of Graph C_m^k(P_2,…,P_2,P_l)

设圈C=v1v2…vmv1,m≥3.在圈C的顶点vi1,vi2,…,vik上分别悬挂k条路Pn1,Pn2,…,Pnk的图记为Ci1,i2,…,ik(Pn1,Pn2,…,Pnk),其中1≤ij≤m,1≤j≤k.在顶点vm上悬挂k条路Pn1,Pn2,…,Pnk的图简记为Cmk(Pn1,Pn2,…,Pnk).利用图Cmk(P2,…,P2,Pl)的特征多项式获得:λ1(Cmk+1(P2,…,P2,Pl-1))≥λ1(Cmk(P2,…,P2,Pl))≥2,其中,k,l∈N,l≥3.

Let C=v1v2…vm,m≥3, be a cycle. A new graph, denoted by Ci1,i2,…ik（Pn1,Pn2,…,Pnk）,1≤ij≤m,1≤j≤k,is obtained by attaching k paths Pn1,Pn2,…,Pnk to the vertices vi1,vi2,…,vik of the cycle c, and Cmk（Pn1,Pn2,…Pnk） is the graph obtained by attaching the paths Pn1,Pn2,…,Pnk to the vertex Vm of the cycle c. In this paper, by using the characteristic polynomial of the graph Cmk（P2,…,P2,P1）, the inequality λ1（Cmk＋（P2,…,P2,Pl-1）≥λ1（Cmk（P2,…,P2,P1））≥2 is obtained, where k,l∈N,l∈3.