69,083 research outputs found
A graph theoretical Poincare-Hopf Theorem
We introduce the index i(v) = 1 - X(S(v)) for critical points of a locally
injective function f on the vertex set V of a simple graph G=(V,E). Here S(v) =
{w in E | (v,w) in E, f(w)-f(v)<0} is the subgraph of the unit sphere at v in
G. It is the exit set of the gradient vector field. We prove that the sum of
i(v) over V is always is equal to the Euler characteristic X(G) of the graph G.
This is a discrete Poincare-Hopf theorem in a discrete Morse setting. It allows
to compute X(G) for large graphs for which other methods become impractical.Comment: 9 figure
Diameters, distortion and eigenvalues
We study the relation between the diameter, the first positive eigenvalue of
the discrete -Laplacian and the -distortion of a finite graph. We
prove an inequality relating these three quantities and apply it to families of
Cayley and Schreier graphs. We also show that the -distortion of Pascal
graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain
estimates for the convergence to zero of the spectral gap as an application of
the main result.Comment: Final version, to appear in the European Journal of Combinatoric
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