GraphRoots.10.nb

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 "In vs 8 we introduce Delta-type boundary conditions. That is the function \
is continuous and the sum of the outwards derivatives is a multiple of f_0.\n\
In vs 9 we will try to introduce symbolic edgelengths in the ",
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Delta[0,0,0,0,0,0,0,0,0]."
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This program gives all roots of a compact graph with rationally dependent \
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 "Run first to initialize. \[LineSeparator]For each graph we have a \
matrix-valued function S(k) defined in many papers. Let ",
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 "since the programs will then give the wrong results. The DeterminantD I get \
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 "returns all the roots (also called eigenfrequencies) of ",
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rationally dependent no solutions will be found. It finds the roots by \
constructing S(k), and then solving  \[Sum](k) = Det S(k) = 0 where \[Sum](k) \
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\[LineSeparator]\[LineSeparator]",
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options are given, for reasons I don\[CloseCurlyQuote]t know. The graph can \
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 StyleBox[" Two free parameters may be used, c1 and c2. \[LineSeparator]To \
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\[CloseCurlyDoubleQuote]Name\[CloseCurlyDoubleQuote], EdgeLabels\[Rule]\
\[CloseCurlyDoubleQuote]EdgeWeight\[CloseCurlyDoubleQuote]]",
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  FontWeight->"Bold"],
 StyleBox[" where mo is your adjacency matrix. The last two arguments are \
optional.\[LineSeparator]The following cell defines three graphs, in the \
variable mo. h is the corresponding Mathematica graph object and will be \
plotted if the cell is run. You should experiment with the cell content.",
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can be found below.\n",
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ambiguity how to define \[Sum](k) as explained in Ref. 4. I call DeterminantD \
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point is whether the eigenfrequencies and their multiplicities are the same \
as found in the literature. I find this to be true in all cases I have \
tested.\[LineSeparator]I have also manually constructed the scattering \
matrices for three graphs in the paper.",
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 StyleBox[ButtonBox["Kennedy, J. B., Kurasov, P., Malenov\[AAcute], G., & \
Mugnolo, D. (2016). On the Spectral Gap of a Quantum Graph. Annales Henri \
Poincar\[EAcute], 17(9), 2439\[Dash]2473.",
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    URL["https://link.springer.com/article/10.1007/s00023-016-0460-2"], None},
  
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  BaseStyle->"Hyperlink",
  ButtonData->{
    URL["https://physics.fjfi.cvut.cz/publications/mf/2009/malenova_thesis.\
pdf"], None},
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  BaseStyle->"Hyperlink",
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ws13/Workshop_Probab_Anal_Geom/Malenova.pdf"], None},
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 StyleBox[ButtonBox["Berkolaiko G. and Liu W. Simplicity of eigenvalues and \
non-vanishing of eigenfunctions of a quantum graph. Journal of Mathematical \
Analysis and Applications, 445(1):803\[Dash]818, 2017.",
  BaseStyle->"Hyperlink",
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    URL["https://www.sciencedirect.com/science/article/pii/S0022247X1630350X"]\
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graphs.",
  BaseStyle->"Hyperlink",
  ButtonData->{
    URL["https://arxiv.org/pdf/1603.07356.pdf"], None},
  ButtonNote->"https://arxiv.org/pdf/1603.07356.pdf"],
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  BaseStyle->"Hyperlink",
  ButtonData->{
    URL["http://kodu.ut.ee/~blixt/Kandarb.pdf"], None},
  ButtonNote->"http://kodu.ut.ee/~blixt/Kandarb.pdf"],
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 StyleBox[ButtonBox["Boris Gutkin and Uzy Smilansky. Can one hear the shape \
of a graph? Journal of Physics A: Mathematical and General, 34(31):6061, \
2001.",
  BaseStyle->"Hyperlink",
  ButtonData->{
    URL["https://iopscience.iop.org/article/10.1088/0305-4470/34/31/301"], 
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Centre for Mathematical Sciences, Lund University, 2007.",
  BaseStyle->"Hyperlink",
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    URL["http://www.maths.lth.se/matematiklth/personal/kurasov/Master/Marlena-\
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