In finite group theory, studying the prime graph of a group has been an important topic for
almost the past half-century. Recently, prime graphs of solvable groups have been characterized
in graph-theoretical terms only; this now allows the study of these graphs without any knowledge
of the group-theoretical background. We approach prime graphs from a linear-algebraic angle
and focus on the class of minimally connected prime graphs introduced in earlier work on the
subject. As our main results, we prove new properties about the adjacency matrices of some
special families of these graphs, focusing on their characteristic polynomials and spectra.
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