The Neighborhood Characteristic Parameter for Graphs

Abstract

Define the neighborhood characteristic of a graph to be $s_1 - s_2 + s_3 - \cdots$, where $s_i$ 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 $i,j \ge 2$, how many $K_{i,j}$ subgraphs there are or, through an Euler-Poincaré-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.