Functional edge labelings: how far a graph is from being magic

A functional edge labeling of a graph labels the vertices bijectively with $1,\dots,n$ and gives each edge $xy$ the value $\varphi(f(x),f(y))$ of a fixed function $\varphi$; total labelings fit the same scheme with $\varphi(f(x),f(xy),f(y))$. The labeling is $\varphi$-magic when all edges get the same value. We measure how far a graph is from being $\varphi$-magic by its \emph{magic angle}, the smallest angle between the vector of edge values and the all-ones vector over all labelings; its tangent is the coefficient of variation of the edge values, and a graph is $\varepsilon$-magic when its magic angle is at most $\varepsilon$. Edge-magic, super edge-magic, edge-antimagic vertex and prime labelings all appear as special cases. For connected graphs we study the sum, the product, the absolute and the oriented difference, the maximum, the minimum, the greatest common divisor, the least common multiple and the total labelings of Kotzig and Rosa. For the sum, the complete graph has the largest magic angle among graphs of order $n$, the angle is bounded below in terms of the degrees and, for regular graphs, of the smallest eigenvalue, and we compute or estimate it for stars, paths and cycles. For the product, the angle of a complete bipartite graph satisfies a spherical Pythagorean theorem. For the oriented difference we characterize the connected graphs that admit edge values with mean zero. For total labelings, the magic angle of every graph is smaller than $\arctan\frac13$, while the magic angle of $K_n$ is bounded away from zero, a quantitative form of the theorem of Kotzig and Rosa.