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{\noindent \bf Abstract:}
We compute the spectra of the adjacency matrices
of the semi-regular polytopes.
A few different techniques are employed:
the most sophisticated,
which relates the 1-skeleton of the polytope to a Cayley graph,
is based on methods akin to those of Lov\'asz and Babai ([L], [B]).
It turns out that the algebraic degree of the eigenvalues is
at most 5, achieved at two 3-dimensional solids.

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{\nobigbf Introduction}

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Important information about a graph is conveyed by its spectrum,
i.e., the set of eigenvalues of its adjacency matrix;
an extensive bibliography can be found in [CDS].
Symmetries of a graph are helpful in computing the spectrum ([PS]).
If the group of symmetries acts transitively on vertices,
Lov\'asz ([L]) and Babai ([B]) show how to 
apply techniques from representation theory
of groups to reduce significantly the algebraic degree of the problem.

In [ST], we computed the spectra of all regular polytopes
(i.e., of the graph formed by their vertices and edges);
somewhat surprisingly, all eigenvalues have algebraic degree at most 3.
The complete list of regular polytopes is known since Schl\"afli
([Sc], [C]).
{\sl Semi-regular polytopes} are
polytopes with regular faces whose isometry group
acts transitively on vertices.
Trivial examples are regular polytopes and,
in three dimensions, prisms and antiprisms.
Nontrivial 3-dimensional semi-regular polytopes
are known as {\sl Archimedean solids}, despite the fact that
the writings by Archimedes on the subject are lost ([C]).
Kepler ([K]) wrote the first available list of Archimedean solids
with a proof its completeness.
The semi-regular polytopes in higher dimensions
were known to Gosset ([Go])
and were extensively studied by Coxeter ([C2], [C3]),
but only recently Blind and Blind ([BB]) showed
that Gosset's list is complete.

In this paper, we compute the spectra of all semi-regular polytopes.
Section 1 contains our results:
characteristic polynomials of the adjacency matrices
are completely factored in $\ZZ[\tau]$, $\tau = (1 + \sc)/2$.
It turns out that the algebraic degree of the eigenvalues is
at most 5, achieved at two 3-dimensional solids.
Section 2 is devoted to the study of some Archimedean solids
whose spectra can be obtained by taking into account
a few planes of symmetry.
Gosset polytopes are described in section 3:
since they have very large groups of symmetry,
the technique used in [ST] for regular polytopes is convenient here.
In section 4, we show how to apply group representations
to compute the spectrum of a Cayley graph
by Lov\'asz's, Babai's and our modified version of their methods.
Chung and Sternberg ([CS]) used similar ideas to compute
the spectra of some regular graphs and of the buckyball molecule,
which corresponds to a weighted version of
$\langle 5, 6, 6 \rangle$ (in the notation of the next section).
Group representation techniques
are used in section 5 to compute the spectra
of the remaining Archimedean solids.
In section 6 we consider three discrete subgroups of $S^3$,
the unit quaternionic sphere.
These come in handy in sections 7 and 8 where we compute the spectra
of the two missing semi-regular polytopes.
We tried to make this paper accessible to readers
with a scant knowledge of the semi-regular polytopes;
most difficulties should be resolved by consulting
the excellent book [C].

