单圈赋权图的特征多项式  

Characteristic Polynomial of Weighted Unicyclic Graphs

作　　者：胡夫涛[1] 孙美钰 于紫嫣 HU Futao;SUN Meiyu;YU Ziyan(School of Mathematical Sciences,Anhui University,He fei 230601,China)

机构地区：[1]安徽大学数学科学学院,合肥230601

出　　处：《合肥学院学报（综合版）》2022年第5期21-26,共6页Journal of Hefei University：Comprehensive ED

基　　金：安徽省自然科学基金项目"图的控制及相关问题研究"(2108085MA02);安徽省高校自然科学基金项目"基于电阻距离的图结构分析"(KJ2020A0001)。

摘　　要：设G=(V,E)是n阶简单无向图。设w:E(G)→P\{0}P是E(G)上的赋权函数,其中P为任意数域。带有赋权函数w的图G称为赋权图,记为(G,w)。赋权图(G,w)的邻接矩阵,记为A(G,w)=(a_(ij))_(n×n),其中当ij∈E(G)时,a_(ij)=w(ij),当ij■E(G)时,a_(ij)=0。赋权图是一般图和符号图等的推广。赋权图(G,w)的特征多项式φ(G,w)=det(I_(n)-A(G,w))。对于单圈赋权图(G,w),本文建立了φ(G,w)和φ(G,|w|)的等式关系。根据这个等式关系,非平衡符号圈C_(n)^(σ)和非平衡太阳图的特征多项式(C_(n)⊙K_(1))^(σ)分别为φ(C_(n)^(σ))=2 cos(n arccosλ/2)+2,φ((C_(n)⊙K_(1))^(σ))=2λ^(n)cos(n arccosλ/2)+2λ^(n)-1。Let G=(V,E)be a simple undirected graph with orde r n.Let w:E(G)→P\{0}is a weighted function on E(G),where P is any field.A graph G with weighted functio n w is called a weighted graph,denoted by(G,w).The adjacent matrix of weighted graph(G,w),denoted by,where a_(ij)=w(ij)if ij∈E(G),a_(ij)=0 if ij■E(G).Weighted graph is the generalization of simple graph and signed graph.The characteristic polynomial of weighted graph(G,w)isφ(G,w)=det(I_(n)-A(G,w)).For weighted unicyclic graph(G,w),we constructed relationship betweenφ(G,w)andφ(G,w).By the above equality,The characteristic polynomials of unbalanced signed cycle C_(n)^(σ) and unbalanced signed sun graph(C_(n)⊙K_(1))^(σ)areφ(C_(n)^(σ))=2 cos(n arccosλ/2)+2 andφ((C_(n)⊙K_(1))^(σ))=2λ^(n)cos(n arccosλ/2)+2λ^(n)-1,respectively.

关 键 词：赋权图 符号图 特征多项式 圈 

分 类 号：O157.5[理学—数学]