The set of graph eigenvalues of the adjacency matrix is called the spectrum of the graph. (But note that in physics, the eigenvalues of the Laplacian matrix of a graph are sometimes known as the graph's spectrum.) The spectrum of a graph with -fold degenerate eigenvalues is commonly denoted (van Dam and Haemers 2003) or (Biggs 1993, p. 8; Buekenhout and Parker 1998).
The product over the elements of the spectrum of a graph is known as the characteristic polynomial of , and is given by the characteristic polynomial of the adjacency matrix of with respect to the variable .
The largest absolute value of a graph's spectrum is known as its spectral radius.
The spectrum of a graph may be computed in the Wolfram Language using Eigenvalues[AdjacencyMatrix[g]]. Precomputed spectra for many named graphs can be obtained using GraphData[graph, "Spectrum"].
A graph whose spectrum consists entirely of integers is known as an integral graph.
The maximum vertex degree of a connected graph is an eigenvalue of iff is a regular graph.
Two nonisomorphic graphs can share the same spectrum. Such graphs are called cospectral. There seems to be no standard name for graphs known to be uniquely determined by their spectra. While they could conceivably be called spectrally unique, the term "determined by spectrum" has been used in practice (van Dam and Haemers 2003).
A software package known as Graffiti, written in the mid-1980s by Siemion Fajtlowicz, has been used generate a thousand conjectures in spectral graph theory (DeLaVina 2005, Roucairol and Cazenave 2024).