P■形图的伴随多项式的分解及其补图的色等价性

The factorizations of adjoint polynomials of graphs of shape as P■ and chromatically equivalence of their complements

设Pn和Cn是具有n个顶点的路和圈,nG表示n个图G的不相交并。令S*r(m+1)+1表示rPm+2的每个分支的一个1度点重迭后得到的图,EESλ表示把Pm的一个1度点与S*r(m+1)+1的r度点重迭后得到的图,可简记为EESλ,δ=(r+1)m+r;设n(≥3)是奇数,λ=n+2-1(n+1)δ,图PESλ表示把2-1(n+1)EESλ的每个分支的r+1度顶点分别与Pn的下标为奇数的2-1(n+1)个顶点重迭后得到的图,运用图的伴随多项式的性质,讨论了图簇EESλ∪rK1、PESλ∪K1和PESλ∪EESλ的伴随多项式的因式分解式,令n=2k-1q-1,λk=(2kq-1)+2k-1qδ,讨论了图簇PESλ和PESλ∪(k-1)K1的伴随多项式的因式分解式,进而证明了这些图的补图的色等价性。

Let Pn be a path with n vertices and let Cn be a cycle with n vertices,and nG be the union of n graphs G without common vertex.We denote by S*r(m+1)+1 the graph consisting of rPm+2 and by coinciding r vertices of degree 1 of rPm+2,Let EESλ be the graph consisting of Pm and S*r(m+1)+1 by coinciding a vertex of degree 1 of Pm with the vertex of degree r of S*r(m+1)+1,can be abbreviated to EESλ,δ=(r+1)m+r;let n(≥3) is an odd number,λ=n+2-1(n+1)δ,Let PESλ be the graph consisting of 2-1(n+1)EESλ and Pn by coinciding the vertex of degree r+1 of every component of 2-1(n+1)EESλ with 2-1(n+1) vertices which subscript be odd of Pn,respectively;By unsing the properties of adjoint polynomials of graphs or even,we discuss the factorizations of adjoint polynomials of graphs EESλ∪rK1 and PESλ∪K1 and PESλ∪EESλ,Let n=2k-1q-1,let λk=(2kq-1)+2k-1qδ,we discuss the factorizations of adjoint polynomials of graphs PESλand PESλ∪(k-1)K1PESλ,further,we prove chromatically equivalence of complements of these graphs.