圈与路相交的图的最大特征值

On the Maximum Eigenvalues of Graphs with Intersectant Cycles and Paths

设圈C=v1v2…vmv1,m≥3.在圈C的顶点vi1,vi2,…,vil分别悬挂一条路Pk1,Pk2,…,Pkl的图记为Ci1i2…il(Pk1,Pk2,…,Pkl),1≤ij≤m,1≤j≤l.顶点vm悬挂l条路Pk1,Pk2,…,Pkl的图简记为Cml(Pk1,Pk2,…,Pkl).在圈C=v1v2…vmv1的顶点i1上悬挂l条路Pk1,Pk2,…,Pkl的图的最大特征值不小于将l条路分别悬挂在l个顶点i1,i2,…,il的图的最大特征值,即1λ(Cil1(Pk1,Pk2,…,Pkl))≥1λ(Ci1i2…il(Pk1,Pk2,…,Pkl)),1≤ij≤m-1,j=1,2,…,l.

Let a cycle be C=v1v2…vmv1,m≥3, attach pendent paths Pk1,Pk2,…,Pkl to the vertices vi1,vi2,…,vil of C, respectively. Record such a graph as Cili2…il（Pk1,Pk2,…,Pkl）,1≤ij≤m,1≤j≤l and for the graph at whose vertice vm there exist l new paths Pk1,Pk2,…,Pkl it is concisely recorded as Cml （ Pk1,Pk2,…,Pkl） Our research finds that for a graph at whose vertice i1 of C=v1v2…vmv1 there exist 1 paths Pk1,Pk2,…,Pkl its maximum eigenvalue will be no less than that of the graph in which 1 pendent paths are attached to 1 verticesi,i2 ,… ,il respectively. That is to say, ）.λ1（Gi1^l（Pk1,Pk2,…,Pkl））≥（Ci1i2…il（Pk1,Pk2,…Pkl）,for1≤i≤m-1,