most recent 30 from http://mathoverflow.net 2013-05-24T00:55:53Z http://mathoverflow.net/feeds/user/29902 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119839/cheegers-inequality-for-graphs-with-multiple-edges-and-loops/119846#119846 Ralf Rueckriemen 2013-01-25T15:57:19Z 2013-01-25T15:57:19Z

<p>As far as I know, Cheeger's inequality does hold for non-simple graphs. If you look at Chung's book 'Spectral graph theory' she first treats simple graphs and then weighted graphs, which are even more general than loops and multiple edges. Her proof seems to hold for weighted graphs as well.</p>

http://mathoverflow.net/questions/119258/examples-on-small-cut-radius-of-totally-convex-set-in-non-negatively-curved-manif/119276#119276 Ralf Rueckriemen 2013-01-18T15:20:36Z 2013-01-18T15:20:36Z

<p>let M be the infinite cylinder ${(x,y,z) | x^2+y^2=1 }$, choose as $C$ a small geodesic disc around some point $p$ and $q$ the point on the opposite site of the circle with the same z coordinate. Then the distance between $q$ and $C$ is realized by two different paths going around the circle in different directions and these paths end in different points in $C$.</p>

http://mathoverflow.net/questions/116501/upper-bound-on-first-non-zero-eigenvalue-of-the-laplacian/116791#116791 Ralf Rueckriemen 2012-12-19T15:41:46Z 2012-12-19T15:41:46Z

<p>Did you look in Isaac Chavel's book 'Eigenvalues in Riemannian geometry'? It contains various bounds for eigenvalues although at a quick glance I didn't see anything for your particular setting.</p>

http://mathoverflow.net/questions/116256/are-there-pairs-of-combinatorial-graphs-that-are-both-isospectral-and-have-the-sa Ralf Rueckriemen 2012-12-13T09:01:10Z 2012-12-13T18:14:32Z

<p>Two graphs are isospectral if the combinatorial Laplacian on them has the same spectrum, equivalently, the adjacency matrix has the same the set of eigenvalues (including multiplicities). Two graphs have the same matroid if they are 2-isomorphic, that means there exists a bijection between their edge sets that preserves cycles. There are plenty of examples of isospectral graphs (that are not isomorphic) and it is fairly easy to draw examples of 2-isomorphic graphs but can both of these phenomena occur at the same time?</p> <p>In general I don't see any reason why no example of this should exist. I know examples of isospectral graphs can be constructed with Seidel switching but this results in graphs with relatively large average degree and usually high connectivity. On the other hand, 3-connected graphs are uniquely determined by their matroid so any potential example cannot be 3-connected.</p> <p>I am interested in connected non-weighted graphs, I don't care whether they are simple (ie have no loops or multiple edges). Thanks</p>

http://mathoverflow.net/questions/116256/are-there-pairs-of-combinatorial-graphs-that-are-both-isospectral-and-have-the-sa/116258#116258 Ralf Rueckriemen 2012-12-13T09:24:05Z 2012-12-13T09:24:05Z

That was quick, thanks. In these examples the matroid is trivial so the better question would be: Are there 2-connected graphs that are isospectral and have the same matroid.