User ralf rueckriemen - MathOverflow 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 Answer by Ralf Rueckriemen for Cheeger's inequality for graphs with multiple edges and loops? 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 Answer by Ralf Rueckriemen for Examples on small cut radius of totally convex set in non-negatively curved manifold 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 Answer by Ralf Rueckriemen for upper bound on first non-zero eigenvalue of the Laplacian 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 are there pairs of combinatorial graphs that are both isospectral and have the same matroid? 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 Comment by Ralf Rueckriemen 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.