Showing papers by "Hong-Jian Lai published in 2000"
TL;DR: It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|≥5; and that every 5-edge
Abstract: Let A be a finite abelian group and G be a digraph. The boundary of a func- tion f : EOGU7!A is a function q f : VOGU7!A given by q fOvUa P e leaving v fOeUˇ P e entering v fOeU. The graph G is A-connected if for every b : VOGU7!A with P vA VOGU bOvUa0, there is a function f : EOGU7!Af0g such that q f a b. In (J. Combi- natorial Theory, Ser. B 56 (1992) 165-182), Jaeger et al showed that every 3-edge- connected graph is A-connected, for every abelian group A with jAjV 6. It is con- jectured that every 3-edge-connected graph is A-connected, for every abelian group A with jAjV 5; and that every 5-edge-connected graph is A-connected, for every abelian group A with jAjV 3. In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A withjAjV 3.
87 citations
Book•
31 Oct 2000
TL;DR: The symbiotic relationship of matrix theory and combinatorics was discussed in this article, where Liu and Lai provided a broad picture of the use of combinatorial ideas in matrix theory.
Abstract: Combinatorics and Matrix Theory have a symbiotic, or mutually beneficial, relationship. This relationship is discussed in my paper The symbiotic relationship of combinatorics and matrix theoryl where I attempted to justify this description. One could say that a more detailed justification was given in my book with H. J. Ryser entitled Combinatorial Matrix Theon? where an attempt was made to give a broad picture of the use of combinatorial ideas in matrix theory and the use of matrix theory in proving theorems which, at least on the surface, are combinatorial in nature. In the book by Liu and Lai, this picture is enlarged and expanded to include recent developments and contributions of Chinese mathematicians, many of which have not been readily available to those of us who are unfamiliar with Chinese journals. Necessarily, there is some overlap with the book Combinatorial Matrix Theory. Some of the additional topics include: spectra of graphs, eulerian graph problems, Shannon capacity, generalized inverses of Boolean matrices, matrix rearrangements, and matrix completions. A topic to which many Chinese mathematicians have made substantial contributions is the combinatorial analysis of powers of nonnegative matrices, and a large chapter is devoted to this topic. This book should be a valuable resource for mathematicians working in the area of combinatorial matrix theory. Richard A. Brualdi University of Wisconsin - Madison 1 Linear Alg. Applies., vols. 162-4, 1992, 65-105 2Camhridge University Press, 1991.
51 citations