图簇G_(j_1j_2…j_t)^(S^* (i))(p,tkm)的伴随多项式的因式分解及色性分析

The Factorization of Adjoint Polynomials of Graphs G_(j_1j_2…j_t)^(s^* (i))(p,tkm) and Chromatically Equivalence Analysis

令S1,k表示k+1个顶点的星,Pm表示m个顶点的路,G是任意的p阶连通图.设V(Pm)={V1,V2,…,Vm-1,Vm}及相应的度序列为(1,2,…,2,1)。SkPm(i+)1表示把kPm的每个分支的第i个顶点Vi分别与星S1,k的k个1度点重迭后得到的图,用GjS1*j2(…i)jt(p,tkm)表示把tSPkm(i+)1的每个分支的k度点分别与图G的顶点uj1,uj2,…,ujt(t≤p)重迭后得到的图,这里p≥1,k≥2,m≥3,1≤i≤m,t≥1.我们通过讨论图簇SkPm(i+)1∪(k-1)K1、S2Pr(mi)+1,SP((2ri)-1)m+1以及GSj1*j2(…i)jt(p,2rmt),GSj1*j2(…ij)t(p,(2r-1)mt)的伴随多项式的因式分解,证明了它们的补图的色等价图的结构定理.推广了张秉儒证明的文[8]中的定理2和定理4。

Let S1,k be the star with k ＋ 1 vertices, and let Pm denote the path with m vertices,let G be an arbitary connected graph. And let V（Pm ）={V1,V2,…,Vm-1,Vm}with corresponding degree sequence （ 1,2,... ,2,1 ）. Denote by Skm＋1^p（i） the graph consisting of kPm and S1,k by coinciding the ith vertex of each component of kPm with k vertices of degree 1 of S1,k respectively. Let Gj1j2…ji^S^＊（i）（p,tkm）denote the graph obtained from tSkm＋1^p（i） and G by coinciding the vertex of degree k of everyone of tSkm＋1^p（i） with vertices uj1,uj2,……,ujt（t≤p） of graph G respectively, where p≥1,k≥2,m ≥3,1≤i≤m,t≥q,By studyying the factorization of adjoint polynomials of the graphs Skm＋1^p（i）U（k-1）k1,U（k-1）K1、S2rm＋1^P（i）,S（2r-1）m＋1^P（i） and Gj1j2…jt^S＊（i）（p,2rmt）,Gj1j2……jt^S＊（i）（2r-1）mt）,the paper proved the structure theorem of the chromatically equivatent graphs of their complements, and improve Theorem 2 and Theorem 4 in Zhang Bing -ru[ 8 ].