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1. Snake Graphs from Triangulated Orbifolds

   https://doi.org/10.3842/SIGMA.2020.138  

   Banaian, Esther; Kelley, Elizabeth (December 2020, Symmetry, Integrability and Geometry: Methods and Applications)

   null (Ed.)

   We give an explicit combinatorial formula for the Laurent expansion of any arc or closed curve on an unpunctured triangulated orbifold. We do this by extending the snake graph construction of Musiker, Schiffler, and Williams to unpunctured orbifolds. In the case of an ordinary arc, this gives a combinatorial proof of positivity to the generalized cluster algebra from this orbifold. 

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2. Snake graphs and continued fractions

   https://doi.org/10.1016/j.ejc.2020.103081  

   Çanakçı, İlke; Schiffler, Ralf (May 2020, European Journal of Combinatorics)
3. Prime Labelings of Snake Graphs

   Bigham, A.; Donovan, E.A.; Pack, J.; Turley, J; Wiglesworth, L.W. (August 2019, PUMP journal of undergraduate research)

   null (Ed.)

   A prime labeling of a graph G with n vertices is a labeling of the vertices with distinct integers from the set {1,2,...,n} such that the labels of any two adjacent vertices are relatively prime. In this paper, we introduce a snake graph, the fused union of identical cycles, and define a consecutive snake prime labeling for this new family of graphs. We characterize some snake graphs that have a consecutive snake prime labeling and then consider a variation of this labeling. 

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4. Higher dimer covers on snake graphs

   https://doi.org/10.5802/alco.464  

   Musiker, Gregg; Ovenhouse, Nicholas; Schiffler, Ralf; Zhang, Sylvester W (January 2026, Algebraic Combinatorics)

   Snake graphs are a class of planar graphs that are important in the theory of cluster algebras. Indeed, the Laurent expansions of the cluster variables in cluster algebras from surfaces are given as weight generating functions for 1-dimer covers (or perfect matchings) of snake graphs. Moreover, the enumeration of 1-dimer covers of snake graphs provides a combinatorial interpretation of continued fractions. In particular, the number of 1-dimer covers of the snake graph $𝒢[a_1,...,a_n]$is the numerator of the continued fraction $[a_1,...,a_n]$. This number is equal to the top left entry of the matrix product $\left(\begin{array}{cc}{a}_1& 1\\ 1& 0\end{array}\right)\cdots \left(\begin{array}{cc}{a}_n& 1\\ 1& 0\end{array}\right)$. In this paper, we give enumerative results on $m$-dimer covers of snake graphs. We show that the number of $m$-dimer covers of the snake graph $𝒢[a_1,...,a_n]$is the top left entry of a product of analogous $(m+1)$-by- $(m+1)$matrices. We discuss how our enumerative results are related to other known combinatorial formulas, and we suggest a generalization of continued fractions based on our methods. These generalized continued fractions provide some interesting open questions and a possibly novel approach towards Hermite’s problem for cubic irrationals. 

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5. Snake Effect: A Novel Haptic Illusion

   https://doi.org/10.1109/TOH.2021.3070277  

   Severgnini, Frederico M.; Martinez, Juan S.; Tan, Hong Z.; Reed, Charlotte M. (October 2021, IEEE Transactions on Haptics)