  This article studies the Lie algebra $Diff(K\Gamma)$ of derivations on the
path algebra $K\Gamma$ of a quiver $\Gamma$ and the Lie algebra on the first
Hochschild cohomology group $H^1(K\Gamma)$. We relate these Lie algebras to the
algebraic and combinatorial properties of the path algebra. Characterizations
of derivations on a path algebra are obtained, leading to a canonical basis of
$Diff(K\Gamma)$ and its Lie algebra properties. Special derivations are
associated to the vertices, arrows and faces of a quiver, and the concepts of a
connected matrix and boundary matrix are introduced to study the relations
among these derivations, concluding that the space of edge derivations is the
direct sum of the spaces of the vertex derivations and the face derivations,
while the dimensions of the latter spaces are the largest possible. By taking
dimensions, this relation among spaces of derivations recovers Euler's
polyhedron formula. This relation also leads a combinatorial construction of a
canonical basis of the Lie algebra $H^1(K\Gamma)$, together with a semidirect
sum decomposition of $H^1(K\Gamma)$.
