  Based on results by Brugall\'e and Mikhalkin, Fomin and Mikhalkin give
formulas for computing classical Severi degrees $N^{d, \delta}$ using long-edge
graphs. In 2012, Block, Colley and Kennedy considered the logarithmic version
of a special function associated to long-edge graphs appeared in
Fomin-Mikhalkin's formula, and conjectured it to be linear. They have since
proved their conjecture. At the same time, motivated by their conjecture, we
consider a special multivariate function associated to long-edge graphs that
generalizes their function. The main result of this paper is that the
multivariate function we define is always linear. A special case of our result
gives an independent proof of Block-Colley-Kennedy's conjecture.
  The first application of our linearity result is that by applying it to
classical Severi degrees, we recover quadraticity of $Q^{d, \delta}$ and a
bound $\delta$ for the threshold of polynomiality of $N^{d, \delta}.$ Next, in
joint work with Osserman, we apply the linearity result to a special family of
toric surfaces and obtain universal polynomial results having connections to
the G\"ottsche-Yau-Zaslow formula. As a result, we provide combinatorial
formulas for the two unidentified power series $B_1(q)$ and $B_2(q)$ appearing
in the G\"ottsche-Yau-Zaslow formula.
  The proof of our linearity result is completely combinatorial. We define
$\tau$-graphs which generalize long-edge graphs, and a closely related family
of combinatorial objects we call $(\tau, n)$-words. By introducing height
functions and a concept of irreducibility, we describe ways to decompose
certain families of $(\tau, n)$-words into irreducible words, which leads to
the desired results.
