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The Connectivity and the Harary Index of a Graph
The Harary index of a graph is defined as the sum of reciprocals of distances
between all pairs of vertices of the graph. In this paper we provide an upper
bound of the Harary index in terms of the vertex or edge connectivity of a
graph. We characterize the unique graph with maximum Harary index among all
graphs with given number of cut vertices or vertex connectivity or edge
connectivity. In addition we also characterize the extremal graphs with the
second maximum Harary index among the graphs with given vertex connectivity
Non-crossing frameworks with non-crossing reciprocals
We study non-crossing frameworks in the plane for which the classical
reciprocal on the dual graph is also non-crossing. We give a complete
description of the self-stresses on non-crossing frameworks whose reciprocals
are non-crossing, in terms of: the types of faces (only pseudo-triangles and
pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a
geometric condition on the stress vectors at some of the vertices.
As in other recent papers where the interplay of non-crossingness and
rigidity of straight-line plane graphs is studied, pseudo-triangulations show
up as objects of special interest. For example, it is known that all planar
Laman circuits can be embedded as a pseudo-triangulation with one non-pointed
vertex. We show that if such an embedding is sufficiently generic, then the
reciprocal is non-crossing and again a pseudo-triangulation embedding of a
planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation
embedding of a planar Laman circuit, the reciprocal is still non-crossing and a
pseudo-triangulation, but its underlying graph may not be a Laman circuit.
Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal
arise as the reciprocals of such, possibly singular, stresses on
pseudo-triangulation embeddings of Laman circuits.
All self-stresses on a planar graph correspond to liftings to piece-wise
linear surfaces in 3-space. We prove characteristic geometric properties of the
lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure
Nonuniform random geometric graphs with location-dependent radii
We propose a distribution-free approach to the study of random geometric
graphs. The distribution of vertices follows a Poisson point process with
intensity function , where , and is a
probability density function on . A vertex located at
connects via directed edges to other vertices that are within a cut-off
distance . We prove strong law results for (i) the critical cut-off
function so that almost surely, the graph does not contain any node with
out-degree zero for sufficiently large and (ii) the maximum and minimum
vertex degrees. We also provide a characterization of the cut-off function for
which the number of nodes with out-degree zero converges in distribution to a
Poisson random variable. We illustrate this result for a class of densities
with compact support that have at most polynomial rates of decay to zero.
Finally, we state a sufficient condition for an enhanced version of the above
graph to be almost surely connected eventually.Comment: Published in at http://dx.doi.org/10.1214/11-AAP823 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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