Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-{kappa}^2$. Using the cone total curvature TC($Gamma$) of a graph $Gamma$ which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface $Sigma$ spanning a graph $Gamma;subset;M$ is less than or equal to $frac{1}{2pi}{TC(Gamma)-{kappa}^2Area(p{ imes}Gamma)}$. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when n = 3, this density estimate implies that if $TC(Gamma)$ < $3.649{pi};+;{kappa}^2inflimits_{p{in}F}Area(p{ imes}{Gamma})$, then the only possible singularities of a piecewise smooth (M, 0, $delta$)-minimizing set $Sigma$ are the Y-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $pi$/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.