树与路的冠图的临界群(英文)  

The Critical Group of Corona of Trees and Paths

作　　者：谭湘花[1] 侯耀平[2] 曾维理[3] 

机构地区：[1]南京工业职业技术学院人文数理系,中国南京210046 [2].湖南师范大学数学与计算机科学学院,中国长沙410081 [3]东南大学交通学院,中国南京210096

出　　处：《湖南师范大学自然科学学报》2012年第3期10-15,共6页Journal of Natural Science of Hunan Normal University

基　　金：国家科技支撑计划资助项目(2009BAG13A06);江苏省普通高校研究生科研创新计划资助项目(CX22-0163);东南大学优秀博士论文基金资助项目(YBJJ1140)

摘　　要：图的临界群是图生成树数目的一个加细.它是图的一个精细不变量,与图的Laplacian矩阵密切相关.将冠图分为点冠图和边冠图,通过在整数环Z上实施一系列的行列变换来计算整数矩阵的Smith标准型,从而确定了点冠图Tm○Pn和边冠图Tm◇Pn的临界群的代数结构.进一步,证明了点冠图Tm○Pn和边冠图Tm◇Pn的临界群的Smith标准型分别为m和2(m-1)个循环群的直和,同时给出了图Tm○Pn和Tm◇Pn的生成树数目.The critical group of a graph is a refinement of the number of spanning trees of the graph.It is a subtle isomorphism invariant of a graph and is closely connected with the graph Laplacian matrix.The corona of a graph is divided into the vertex corona and the edge corona.Through the implementation of a series of row and column operations in the ring Z of integers,the Smith normal form of an integer matrix is obtained.Hence,the structures of the critical group on the vertex corona Tm○Pn and the edge corona Tm◇Pn are determined.Furthermore,it is proved that the Smith normal forms of critical group of Tm○Pn and Tm◇Pn are the direct sum of m and 2（m-1） cyclic groups,respectively.At the same time the number of spanning trees in Tm○Pn and Tm◇Pn are also given.

关 键 词：LAPLACIAN矩阵 临界群 Smith标准型 点冠图 边冠图 

分 类 号：O157.5[理学—数学]