Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The Neighborhood Characteristic Parameter for Graphs | The Neighborhood Characteristic Parameter for Graphs Terry A. McKee Department of Mathematics Statistics Wright State University Dayton Ohio 45435 USA Submitted Aug 19 2001 Accepted May 5 2003 Published May 7 2003 MR Subject Classifications 05C75 05C99 Abstract Define the neighborhood characteristic of a graph to be Si s2 S3 where si counts subsets of i vertices that are all adjacent to some vertex outside the subset. This amounts to replacing cliques by neighborhoods in the traditional Euler characteristic the number of vertices minus the number of edges plus the number of triangles etc. . The neighborhood characteristic can also be calculated by knowing for all i j 2 how many Kij subgraphs there are or through an Euler-Poincare-type theorem by knowing how those subgraphs are arranged. Chordal bipartite graphs are precisely the graphs for which every nontrivial connected induced subgraph has neighborhood characteristic 2. 1 The Neighborhood Characteristic Define the neighborhood characteristic of any graph G without isolated vertices to be vchar G Si S2 S3 1 where si is the number of subsets of V G of cardinality i that are externally dominated meaning that S c N v for some v E V G S. Thus si n is just the order of G and s2 is the number of pairs of vertices that have a common neighbor. For comparison the traditional Euler characteristic 7 which might be thought of as the clique characteristic is char G ki k2 k3 2 where ki is the number of complete subgraphs of G of order i thus ki si k2 m is the number of edges and k3 is the number of triangles. So vchar G can be thought of as modifying char G by replacing complete subgraphs with externally dominated subgraphs. In THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R20 1 topological terms char G is the characteristic of the simplicial complex whose simplices are the complete subgraphs of G and Nchar G is the characteristic of the neighborhood complex N G as in 1 whose simplices are the .
