Chris Godsil
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 6h comment Automorphisms of Graphs If the isomorphism does not extend, the vertices are said to pseudo-similar, and there is some literature. For fine regular graphs, the isomorphism always extends. I do not recall any signs of $\mathbb{Z}_2$ in this situation. Nov 23 comment If $G$ has central Sylow $2$-subgroup, then it has normal $2$-complement. Or you could prove the result when the Sylow 2-subgroup has order two, and use induction. Nov 22 comment Finding all eigenvalues of the adjacency matrix of a simple graph What I use is the theory of equitable partitions, but there isn't room to treat that here. Nov 22 comment Finding all eigenvalues of the adjacency matrix of a simple graph The eigenvectors that are zero on $w$, $x$, $y$ and $z$ form a subspace with dimension $m+n-2$ and all these vectors have eigenvalue $-1$. Any vector that is zero on the four middle vertices and sums to zero on vertices in $G\setminus w$ and $H\setminus z$ will be an eigenvector. Nov 22 comment Finding all eigenvalues of the adjacency matrix of a simple graph There will be a space of dimension $m+n-2$, spanned by eigenvectors with eigenvalue $-1$ that take the value zero on $w$, $x$, $y$ and $z$. This gives a factor of $(x+1)^{m+n-2}$ in the characteristic polynomial. The remaining six eigenvalues are zeros of a degree six polynomial with coefficients depending on $m$ and $n$. In general (if $m\ne n$) there will be no useful formula for the zeros. If $m=n$ this polynomial will factor into two cubics, and there will be very messy formulas. Nov 19 comment Proof of existence of simple group of Order 168 in Dummit and Foote I think you mean 168. Nov 16 comment Why is the commutator expressed as $aba^{-1}b^{-1}$ instead of $a^{-1}b^{-1}ab$? I think it is very common for group theorists to use the $a^{-1}b^{-1}ab$ version. Nov 16 answered Significance of the graph represented by the inverse of the adjacency matrix of another graph Nov 12 answered Proof that Laplacian spectrum is symmetric for bipartite graphs Nov 11 answered For what integers n is the group S_n generated by an element of order 2 together with an element of order 3? Nov 9 comment Generating random groups satisfying certain conditions I am minded of Gordon Royle's quote: sometimes a couple of days enthusiatic programming can save you an hour's careful thought. Nov 7 answered What are the conjugacy class and homotopy class of fundamental group in graph theory? Nov 5 answered Non isomorphic graph and spectrum of a adjacency matrix Oct 30 comment Chromatic number of complement of Petersen graph The complement of the Petersen graph is the line graph of $K_5$, so you want the edge chromatic numer of $K_5$. Oct 27 revised How to solve this graphical function transfromation problem? fixed tag Oct 27 comment Prove that the number of conjugacy classes of a finite group $G$ is given by $k(G) = \frac{1}{|G|}\sum_{g \in G}{\left|C(g)\right|}$ This is also an easy consequence of Burnside's theorem, since the number of conjugacy classes is the number of orbits of $G$ acting on itself by conjugation, and the set of points fixed by an element $g$ of $G$ is equal to $C_G(g)$. Oct 14 comment How can I prove the assertion for a Graph G? The sum of the eigenvalues of $A$ is zero. Oct 13 comment How can I prove the assertion for a Graph G? The least eigenvalue cannot be positive. Oct 9 revised Plotting points of Musical Note frequencies linearly on a Graph fixed tags Oct 5 reviewed Close How do you resist thinking about you may not be able to go further in mathematics?