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2. Hearing the Shape of a Graph

Can one distinguish graphs by listening to them? This is a question which has intrigued me since I have been investigating an area of mathematics known as Spectral Graph Theory, wherein one of the central questions is whether a graph can be distinguished by certain characteristic values called eigenvalues which are associated with the graph. Here are several examples of "chords of eigenvalues" which are associated with various graphs. Be sure to listen carefully for which chords are harmonious and which are dissonant. There is a tie between symmetry in the graph and harmoniousness of its chord!

How an eigenvalue chord is played: values between 0 and 2 are place on two octaves starting at A (440 Hz). Thus a value of 1 corresponds to A (880 Hz), a value of 2 corresponds to A (1760 Hz), a value of 1.5 corresponds to D# (660 Hz), and so forth. All the notes are played simultaneously in a chord. If a graph has a multiple eigenvalue, the corresponding note is played louder.

2x2 grid graph Figure 1 to the left depicts a graph which is a 2 by 2 grid. This graph is very symmetric - just by looking at it you can see the horizontal, vertical, and diagonal symmetries of the graph. As it happens, the eigenvalues of the graph, displayed in Figure 2, also have a lot of symmetry. The horizontal axis gives the values of the eigenvalues, and the vertical axis gives an idea of which eigenvalues appear more than once, by the height of the peak. The eigenvalues are 0, 1, 1, and 2.
Listen to 2x2 grid graph chord (.wav file) 2x2
spectrum

The second graph, depicted in Figure 3, is the first graph with one edge deleted. you might suspect that this disrupts the harmony of the graph's chord. Indeed, it does disrupt it to some extent, but there is still enough symmetry in the graph that the chord is reasonably harmonious. The result of deleting the edge is that the two eigenvalues that corresponded to A (880 Hz) are split into one at D# (660 Hz) and another an octave higher at D# (1320 Hz).
Listen to 2nd graph chord (.wav file) 2x2 spectrum

For the following graphs, click on them to hear their chords.

Listen carefully to the chords of graphs that differ from each other by only one or two edges. Try to distinguish which notes in the chord are new or missing. Listen to the chord and decide if its harmoniousness or dissonance seems to correspond to the visual symmetry of the graph.