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 notation 
is also used for graph eigenvalues (e.g., Godsil
and Royle 2001, p. 193), though sometimes with an indexing convention starting
with 
being the smallest (instead of 
). A graph with diameter 
has a spectrum consisting of at least
distinct eigenvalues, with the bound
becoming tight for a distance-regular graph,
which has exactly 
distinct eigenvalues (Fallat et al. 2024).
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).
." Disc. Math. 186, 69-94, 1998.Cvetković,
D. M.; Doob, M.; and Sachs, H.