The critical group C（G） of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r（G） be the minimum number of generators （i.e., the rank） of the group C（G） and β（G） be the number of independent cycles of G. In this paper, some forbidden induced subgraphs axe given for r（G） = n - 3 and all graphs with r（G） = j3（G） = n - 3 are characterized
In this paper, we introduce the m-Cartan matrix and observe that some properties of the quadratic form associated to the Cartan matrix of an Euclidean diagram can be generalized to the m-Cartan matrix of a McKay quiver. We also describe the McKay quiver for a finite abelian subgroup of a special linear group.
GUO JinYun School of Mathematics and Computational Sciences, Xiangtan University, Xiangtan 411105, China
In this paper the sufficient and necessary conditions are given for a formal triangular matrix ring to be right PP, generalized right PP, or semihereditary, respectively.
