Signed a-polynomials of graphs and Poincar\'e polynomials of real toric manifolds
Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 467-481
https://doi.org/10.4134/BKMS.2015.52.2.467
Published online March 31, 2015
Seunghyun Seo and Heesung Shin
Kangwon National University, Inha University
Abstract : Choi and Park introduced an invariant of a finite simple graph, called \emph{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 \emph{signed a-polynomial} which is a generalization of the signed a-number and gives \emph{a-, b-, and c-numbers}. The signed a-poly\-nomial of a graph $G$ is related to the Poincar\'e polynomial $P_{M(G)}(z)$, 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 multipartite graphs $G$.
Keywords : graph invariant, toric topology, Poincar\'e polynomial
MSC numbers : 05A15, 05C30, 37F20
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