# Hearing Graphs

In discrete mathematics, a graph is a set of unlabeled vertices (points) where each pair of vertices is either connected by an edge (a line segment) or not. A graph is therefore an abstraction of the concept of a network of objects. Graphs have been intensively studied in a field known as graph theory.

The *adjacency matrix* of a graph is a two-dimensional grid of integers describing the connections between nodes. A subfield of graph theory called *spectral graph theory* concerns the eigenvalues of this adjacency matrix, known as the graph spectrum (as well as other matrices associated with the graph). A graph's spectrum is an example of a graph invariant; i.e. it doesn't depend on any arbitrary factor such as the order of the nodes in the graph.

The spectrum of a graph is a multiset of real numbers. It can therefore can be sonified by treating the eigenvalues as frequency ratios. Here, we play the eigenvalues with piano samples in ascending order with frequencies relative to middle C. Zero frequency is inaudible, so zero eigenvalues are represented using a bass drum sample, and negative eigenvalues are represented with time-reversed piano samples. *Hearing Graphs* uses this method to sonify all spectrally distinct graphs with vertex count up to 6.