Asymptotics of the Euler number of bipartite graphs

Details Asymptotics of the Euler number of bipartite graphs Asymptotics of the Euler number of bipartite graphs

2007-04-13

arxiv.org/abs/0704.1782v1

Advances in Applied Mathematics 44 (2010) 155--167

Mathematics

Combinatorics

13 pages, 6 figure, submitted to JCTB

Scientific paper

10.1016/j.aam.2009.05.002

We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing the Euler number of certain subgraphs of the Cartesian product of a graph $G$ with the path $P_m$ in terms of self adjoint operators. The asymptotic expansion of the Euler number is given in terms of the eigenvalues of the associated operator. For two classes of graphs, the comb graphs and the Cartesian product $P_2 \Box P_m$, we numerically solve the eigenvalue problem.

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