  Binary relations are one of the standard ways to encode, characterise and
reason about graphs. Relation algebras provide equational axioms for a large
fragment of the calculus of binary relations. Although relations are standard
tools in many areas of mathematics and computing, researchers usually fall back
to point-wise reasoning when it comes to arguments about paths in a graph. We
present a purely algebraic way to specify different kinds of paths in relation
algebras. We study the relationship between paths with a designated root vertex
and paths without such a vertex. Since we stay in first-order logic this
development helps with mechanising proofs.To demonstrate the applicability of
the algebraic framework we verify the correctness of three basic graph
algorithms. All results of this paper are formally verified using the
interactive proof assistant Isabelle/HOL.
