湖南师范大学自然科学学报
湖南師範大學自然科學學報
호남사범대학자연과학학보
ACTA SCIENTIARUM NATURALIUM UNIVERSITATIS NORMALIS HUNANENSIS
2012年
3期
10-15
,共6页
Laplacian矩阵%临界群%Smith标准型%点冠图%边冠图
Laplacian矩陣%臨界群%Smith標準型%點冠圖%邊冠圖
Laplacian구진%림계군%Smith표준형%점관도%변관도
Laplacian matrix%critical group%the Smith normal form%the vertex corona%the edge corona
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.
圖的臨界群是圖生成樹數目的一箇加細.它是圖的一箇精細不變量,與圖的Laplacian矩陣密切相關.將冠圖分為點冠圖和邊冠圖,通過在整數環Z上實施一繫列的行列變換來計算整數矩陣的Smith標準型,從而確定瞭點冠圖Tm○Pn和邊冠圖Tm◇Pn的臨界群的代數結構.進一步,證明瞭點冠圖Tm○Pn和邊冠圖Tm◇Pn的臨界群的Smith標準型分彆為m和2(m-1)箇循環群的直和,同時給齣瞭圖Tm○Pn和Tm◇Pn的生成樹數目.
도적림계군시도생성수수목적일개가세.타시도적일개정세불변량,여도적Laplacian구진밀절상관.장관도분위점관도화변관도,통과재정수배Z상실시일계렬적행렬변환래계산정수구진적Smith표준형,종이학정료점관도Tm○Pn화변관도Tm◇Pn적림계군적대수결구.진일보,증명료점관도Tm○Pn화변관도Tm◇Pn적림계군적Smith표준형분별위m화2(m-1)개순배군적직화,동시급출료도Tm○Pn화Tm◇Pn적생성수수목.
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.
