令G=(V(G),E(G))是n个点、m条边的简单图,σ:E(G)→{+1,-1}是定义在边集E(G)上的符号映射,称Γ=(G,σ)为G的一个符号图.给定一个符号图Γ,Belardo和Simic'定义了符号线图(Γ)和符号剖分图S(Γ),并得到它们邻接特征多项式和Γ的Laplacian特征多项式之间的关系.本文定义了另外三类符号变换图,即符号中间图、符号三角扩展图和符号全图,分别记为Q(Γ)、R(Γ)和T(Γ).当G是正则图,给出这三类符号变换图的邻接特征多项式和Laplacian特征多项式与原符号图对应多项式的关系.这些结果推广了一般图对应的已有结论.
Let G=(V(G),E(G)) be a simple graph of order n and size m,and let σ:E(G)→{+1,-1} be a mapping defined on the edges of G. We call Γ=(G,σ)a signed graph of G.Given a signed graph Γ,Belardo and Simic' defined the signed line graph (Γ) as well as the signed subdivision graph S(Γ) and obtained relations between their adjacency polynomials and Laplacian polynomials of Γ.In this paper, we defined other three classes of signed transform graphs,signed middle graph,signed triangular extension graph and signed total graph,denoted by Q(Γ),R(Γ),T(Γ)respectively.When G is regular,we express the adjacency and Laplacian polynomials of the three classes of signed transform graphs in terms of those of original signed graph Γ.These results generalize the counterpart of unsigned graphs.
