• 1 Department of Mathematics [Berkeley]

We construct and study an embedded weighted balanced graph in $\mathbb{R}^{n+1}$ parametrized by a strictly increasing sequence of $n$ coprime numbers $\{ i_1, \ldots, i_n\}$, called the $\textit{tropical secant surface graph}$. We identify it with the tropicalization of a surface in $\mathbb{C}^{n+1}$ parametrized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector $(0, i_1, \ldots, i_n)$, which can be described by a balanced graph called the $\textit{tropical secant graph}$. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in $\mathbb{P}^4$.