SIGNED A-POLYNOMIALS OF GRAPHS AND POINCAR´E POLYNOMIALS OF REAL TORIC MANIFOLDS
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Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-poly- nomial of a graph G is related to the Poincar´e polynomial PM(G)(z), which is the generating function for the Betti numbers of the real toric manifold M(G). We give the generating functions for the signed a- polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold M(G) for complete multi- partite graphs G.
Abstract
Choi and Park introduced an invariant of a finite simple graph, called signed a-number, arising from computing certain topological invariants of some specific kinds of real toric manifolds. They also found the signed a-numbers of path graphs, cycle graphs, complete graphs, and star graphs. We introduce a signed a-polynomial which is a generalization of the signed a-number and gives a-, b-, and c-numbers. The signed a-poly- nomial of a graph G is related to the Poincar´e polynomial PM(G)(z), which is the generating function for the Betti numbers of the real toric manifold M(G). We give the generating functions for the signed a- polynomials of not only path graphs, cycle graphs, complete graphs, and star graphs, but also complete bipartite graphs and complete multipartite graphs. As a consequence, we find the Euler characteristic number and the Betti numbers of the real toric manifold M(G) for complete multi- partite graphs G.
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