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 Kleene relation algebras, which are relation algebras equipped with an operation for reflexive transitive closure. 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.

二进制关系是对图进行编码，表征和推理的标准方法之一。关系代数为二元关系演算的很大一部分提供方程式公理。尽管关系是数学和计算的许多领域中的标准工具，但是当涉及到有关图形中路径的论证时，研究人员通常会退回到逐点推理。我们提出了一种纯代数方式，以在Kleene关系代数中指定不同种类的路径，这些关系代数配备了自反传递闭合的运算。我们研究具有指定根顶点的路径与没有该顶点的路径之间的关系。由于我们始终采用一阶逻辑，因此这一发展有助于机械化证明。为了证明代数框架的适用性，我们验证了三种基本图形算法的正确性。本文的所有结果均使用交互式校对助手Isabelle / HOL进行了正式验证。