The Neighborhood Characteristic Parameter for Graphs

Define the neighborhood characteristic of a graph to be s1 − 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 Ki,j subgraphs there are or, through an Euler-Poincarétype 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 Nchar(G) = s1 − s2 + s3 − · · · , (1) where si is the number of subsets of V (G) of cardinality i that are externally dominated, meaning that S ⊆ N(v) for some v ∈ V (G) − S. Thus s1 = 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) = k1 − k2 + k3 − · · · , (2) where ki is the number of complete subgraphs of G of order i; thus k1 = s1, k2 = m is the number of edges, and k3 is the number of triangles. So Nchar(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 externally dominated subgraphs of G.