# Snake Graphs Arising from Groves with an Application in Coding Theory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivations

#### 1.2. Contributions

## 2. Background and Related Work

#### 2.1. Path Algebras

#### 2.2. Snake Graphs

**Remark**

**1.**

#### 2.3. String Modules and Snake Graphs

- If $w=\u2300$, then the corresponding abstract snake graph is given by a single tile.
- If there is at least one letter, then ${a}_{1},{a}_{2},\dots ,{a}_{n}$ is a concatenation of a collection of alternating maximal direct and inverse strings ${w}_{i}$ such that $w={w}_{1}{w}_{2}\dots {w}_{k}$. Each ${w}_{i}$ might be of length 1.
- For each ${w}_{i}$, we construct a zigzag snake graph ${\mathcal{G}}_{i}$ with $l\left({w}_{i}\right)+1$ tiles, where $l\left({w}_{i}\right)$ is the number of direct or inverse arrows in ${w}_{i}$. Let ${\mathcal{G}}_{i}$ be the zigzag snake graph with tiles ${T}_{1}^{i},\dots ,{T}_{l\left({w}_{i}\right)+1}^{i}$, such that ${T}_{2}^{i}$ is glued to the right (resp. on top) of ${T}_{1}^{i}$ if ${w}_{i}$ is direct (resp. inverse).
- We now glue ${\mathcal{G}}_{i+1}$ to ${\mathcal{G}}_{i}$, for all i, by identifying the last tile ${T}_{l\left({w}_{i}\right)+1}^{i}$ of ${\mathcal{G}}_{i}$ and the first tile ${T}_{1}^{i+1}$ of ${\mathcal{G}}_{i+1}$, such that ${T}_{l\left({w}_{i}\right)}^{i}$, ${T}_{l\left({w}_{i}\right)+1}^{i}$, ${T}_{2}^{i+1}$ is a straight piece.

- $1\u27f62\u27f63\u27f64$, which defines the zigzag snake graph ${\mathcal{G}}_{1}$ containing the tiles, $1,2,3,$ and 4.
- $4\u27f55$, which defines the straight snake graph ${\mathcal{G}}_{2}$ containing the tiles 4 and 5.

#### 2.4. Brauer Configuration Algebras

- ${\Gamma}_{0}$ is a finite set of vertices.
- ${\Gamma}_{1}$ is a collection of polygons, which are labeled multisets consisting of vertices (vertices repetition allowed). Each polygon contains more than one vertex.
- $\mu $ is a map from the set of vertices ${\Gamma}_{0}$ to the set of positive integers $\mathbb{N}\backslash \left\{0\right\}={\mathbb{N}}^{+}$, $\mu :{\Gamma}_{0}\to {\mathbb{N}}^{+}.$
- $\mathcal{O}$ is a choice for each vertex $\alpha \in {\Gamma}_{0}$, of a cyclic ordering of the polygons in which $\alpha $ occurs as a vertex including repetitions (see [9] for more details). For instance, if a vertex $\alpha \in {\Gamma}_{0}$ occurs in polygons ${U}_{{i}_{i}},{U}_{{i}_{2}},\dots ,{U}_{{i}_{m}}$, for suitable indices ${i}_{1},{i}_{2},\dots ,{i}_{m}\in \{1,2,3,\dots ,n\}$, then the cyclic order is obtained by linearly ordering the list, say$${U}_{{i}_{1}}^{{\alpha}_{1}}<{U}_{{i}_{2}}^{{\alpha}_{2}}<\cdots <{U}_{{i}_{m}}^{{\alpha}_{m}},\phantom{\rule{1.em}{0ex}}{\alpha}_{{i}_{s}}>0.$$The sequence (2) is said to be the successor sequence at vertex $\alpha $ denoted ${S}_{\alpha}$, which is unique up to permutations.Henceforth, this paper assumes the notation used in [12] for successor sequences and special cycles. Namely, if a vertex ${\alpha}^{\prime}\ne \alpha $ belongs to some polygons ${U}_{{j}_{1}},{U}_{{j}_{2}},\dots {U}_{{j}_{k}}$ ordered according to the already defined cyclic ordering associated with the vertex $\alpha $, then we will assume that, up to permutations, the cyclic ordering associated with the vertex ${\alpha}^{\prime}$ is built, taking into account that polygons ${U}_{{j}_{1}},{U}_{{j}_{2}},\dots {U}_{{j}_{k}}$ inherit the order given by the successor sequence ${S}_{\alpha}$.

- Identify special cycles associated with non-truncated vertices in the same polygon (i.e., if ${\delta}_{1},{\delta}_{2}\in U$ with $U\in {\Gamma}_{1}$, then ${C}_{{\delta}_{1}}^{\mu \left({\delta}_{1}\right)}-{C}_{{\delta}_{2}}^{\mu \left({\delta}_{2}\right)}\in {\rho}_{\Gamma}$).
- If ${C}_{\delta}$ is a special cycle associated with a non-truncated vertex $\delta $, then a product of the form ${C}^{\mu \left(\delta \right)}a\in {\rho}_{\Gamma}$, if a is the first arrow of $\delta $.
- Quadratic monomial relations of the form $ab$ in $\mathbb{F}{Q}_{\Gamma}$, where $ab$ is not a subpath of any special cycle unless $a=b$ and a is a loop associated with a vertex $\alpha $ of valency 1 and $\mu \left(\alpha \right)>1$.

**Theorem**

**1**

- 1.
- There is a bijection between ${\Gamma}_{1}$ and the set of indecomposable projective Λ-modules.
- 2.
- If P is a projective indecomposable Λ-module corresponding to a polygon V in Γ, then $\mathrm{rad}\left(P\right)$ is a sum of r indecomposable uniserial modules, where r is the number of (non-truncated) vertices of V and where the intersection of any two of the uniserial modules is a simple Λ-module.

**Proposition**

**1**

**Theorem**

**2**

## 3. Main Results

#### 3.1. Admissible Words

- • is the usual concatenation of words. If no confusion arises, later on, we will write $w\u2022{w}^{\prime}=w{w}^{\prime}$.
- If, for $n\ge 1$ fixed $w\in {\mathcal{W}}_{n}$, then $\perp \left(w\right)=\left(\right|w|+1)w\in {\mathcal{W}}_{n+1}$. Moreover, $\top \left(w\right)=w\left(\right|w|+1)\in {\mathcal{W}}_{n+1}$.
- If $w,{w}^{\prime}\in \mathcal{W}$, then $w\propto {w}^{\prime}=w\left(\right|w{w}^{\prime}|+1){w}^{\prime}$. In such a case, we write ${(w\propto {w}^{\prime})}^{l}=w$, ${(w\propto {w}^{\prime})}^{r}={w}^{\prime}$. In particular, $\perp \left(w\right)=\u2300\propto w=\left(\right|w|+1)w$, $\top \left(w\right)=w\propto \u2300=w\left(\right|w|+1)$. Thus, any admissible word w can be written in the form $w={\left(w\right)}^{l}\propto {\left(w\right)}^{r}$.
- If $w,{w}^{\prime}\in \mathcal{W}$, then $w\cup {w}^{\prime}=\{w,{w}^{\prime}\}$. If ${w}_{1},{w}_{2},\dots ,{w}_{k}\in \mathcal{W}$, then $\perp \left({\displaystyle \bigcup _{k=1}^{n}}{w}_{k}\right)={\displaystyle \underset{k=1}{\bigcup ^{n}}}\perp \left({w}_{k}\right)$. Moreover, $\top \left({\displaystyle \underset{k=1}{\bigcup ^{n}}}{w}_{k}\right)={\displaystyle \underset{k=1}{\bigcup ^{n}}}\top \left({w}_{k}\right)$.
- $w\leftrightarrow {w}^{\prime}=(w\to {w}^{\prime})\cup (w\leftarrow {w}^{\prime})$, where $w\to {w}^{\prime}={w}^{l}\propto ({w}^{r}\leftrightarrow {w}^{\prime})$, and $w\leftarrow {w}^{\prime}=(w\leftrightarrow {\left({w}^{\prime}\right)}^{l})\propto {\left({w}^{\prime}\right)}^{r}$. In particular, if $w=x\left(\right|w\left|\right)$ (${w}^{\prime}=\left(\right|{w}^{\prime}\left|y\right)$), then $w\to {w}^{\prime}=x\left(\right|w{w}^{\prime}\left|\right){w}^{\prime}$ ($w\leftarrow {w}^{\prime}=w\left(\right|w{w}^{\prime}\left|\right)y$).
- $\underset{k=1}{\bigcup ^{n}}}{w}_{k}\to w={\displaystyle \underset{k=1}{\bigcup ^{n}}}{w}_{k}\to w$, $w\leftarrow {\displaystyle \underset{k=1}{\bigcup ^{n}}}{w}_{k}={\displaystyle \underset{k=1}{\bigcup ^{n}}}w\leftarrow {w}_{k}$.

**Proposition**

**2.**

**Proof.**

#### 3.2. Brauer Configuration Algebras Associated with Snake Graphs

**Proposition**

**3.**

**Proof.**(Induction) If ${d}_{n}=1{a}_{1}1{a}_{2}1\dots 1{a}_{n}1$ is the dimension of a string module associated with an orientation ${a}_{1}{a}_{2}\dots {a}_{n}$ for which ${a}_{i}\in \{\to ,\leftarrow \}$, then, according to the operations defined in Section 3.1, ${d}_{n}=21{a}_{2}1\dots 1{a}_{n}1$ if ${a}_{1}=\to $, ${d}_{n}=12{a}_{2}1\dots 1{a}_{n}1$, if ${a}_{1}=\leftarrow $, ${d}_{n}=\perp (12\cup 21){a}_{3}1\dots 1{a}_{n}1=312{a}_{3}1\dots 1{a}_{n}1\cup 321{a}_{3}1\dots 1{a}_{n}1$, if ${a}_{2}=\to $ has 21 as a source. ${d}_{n}=131{a}_{3}1\dots 1{a}_{n}1$ if ${a}_{2}=\to $ has 12 as a source.

**Corollary**

**1.**

**Proof.**(Induction) Note that $|{\mathcal{W}}_{1}|=1$, $|{\mathcal{W}}_{2}|=2$, $|{\mathcal{W}}_{3}|=5$. If it is assumed that the statement is true for $1\le j<i$, then $|{\mathcal{W}}_{i}|=|\perp \left({\mathcal{W}}_{i-1}\right)|+|{\mathcal{W}}_{i-2}\propto {\mathcal{W}}_{1}|+\dots +|{\mathcal{W}}_{i-h}\propto {\mathcal{W}}_{h-1}|+\dots |\top \left({\mathcal{W}}_{i-1}\right)|$=2$\underset{k=0}{\sum ^{\lfloor \frac{i-1}{2}\rfloor}}}{C}_{k}{C}_{i-1-k}={C}_{i$. □

**Theorem**

**3.**

- 1.
- If ${P}_{i}$ is an indecomposable projective module over ${\Lambda}_{{\Gamma}^{n}}$ associated with the polygon ${U}_{i}$. Then, the number of summands in $\mathrm{rad}\phantom{\rule{2.84544pt}{0ex}}{P}_{i}$ is i, $1\le i\le n$.
- 2.
- $val\left(i\right)={C}_{i}{C}_{n-i}(n-i+1)={e}_{in}$,
- 3.
- ${\mathrm{dim}}_{\mathbb{F}}\phantom{\rule{2.84544pt}{0ex}}{\Lambda}^{n}=2({C}_{n}+\underset{i=1}{\sum ^{n}}{t}_{{e}_{in}-1}$),
- 4.
- ${\mathrm{dim}}_{\mathbb{F}}\phantom{\rule{2.84544pt}{0ex}}Z\left({\Lambda}^{n}\right)=1+{C}_{n}+\underset{j=1}{\sum ^{n}}{\nu}_{ij}^{n}$.

**Proof.**

#### 3.3. The Associated Code

**Theorem**

**4.**

**Proof.**

**Corollary**

**2.**

**Proof.**

## 4. Concluding Remarks and Future Work

#### Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

${C}_{j}$ | (jth Catalan number) |

${\mathrm{dim}}_{\mathbb{F}}\phantom{\rule{2.84544pt}{0ex}}{\Lambda}_{\Gamma}$ | (Dimension of a Brauer configuration algebra) |

${\mathrm{dim}}_{\mathbb{F}}\phantom{\rule{2.84544pt}{0ex}}Z\left({\Lambda}_{\Gamma}\right)$ | (Dimension of the center of a Brauer configuration algebra) |

$E(1,{C}_{{\Gamma}^{j}},{h}_{\alpha})$ | (Energy of a code ${C}_{{\Gamma}^{j}}$) |

$\mathbb{F}$ | (Field) |

${\Gamma}_{0}$ | (Set of vertices of a Brauer configuration $\Gamma $) |

${t}_{i}$ | (ith triangular number) |

$\mathrm{occ}(\alpha ,V)$ | (Number of occurrences of a vertex $\alpha $ in a polygon V) |

$w\left(V\right)$ | (The word associated with a polygon V) |

${V}_{i}^{\left(\alpha \right)}$ | (Ordered sequence of polygons) |

$val\left(\alpha \right)$ | (Valency of a vertex $\alpha $) |

## Appendix A. Python Routines

## References

- Propp, J. The combinatorics of frieze patterns and Markoff numbers. Integers
**2020**, 20, 1–38. [Google Scholar] - Çanakçi, I.; Schiffler, R. Cluster algebras and continued fractions. Compos. Math.
**2018**, 54, 565–593. [Google Scholar] [CrossRef] [Green Version] - Çanakçi, I.; Schiffler, R. Snake graphs and continued fractions. Eur. J. Combin.
**2020**, 86, 1–19. [Google Scholar] [CrossRef] [Green Version] - Çanakçi, I.; Schiffler, R. Snake graphs calculus and cluster algebras from surfaces. J. Algebra
**2013**, 382, 240–281. [Google Scholar] [CrossRef] - Çanakçi, I.; Schiffler, R. Snake graphs calculus and cluster algebras from surfaces II: Self-crossings snake graphs. Math. Z.
**2015**, 281, 55–102. [Google Scholar] [CrossRef] [Green Version] - Çanakçi, I.; Schiffler, R. Snake graphs calculus and cluster algebras from surfaces III: Band graphs and snake rings. Int. Math. Res. Not. IMRN
**2017**, rnx157, 1–82. [Google Scholar] [CrossRef] - Musiker, G.; Schiffler, R.; Williams, L. Posiivity for cluster algebras from surfaces. Adv. Math.
**2011**, 227, 2241–2308. [Google Scholar] [CrossRef] [Green Version] - Çanakçi, I.; Schroll, S. Lattice bijections for string modules snake graphs and the weak Bruhat order. Adv. Appl. Math.
**2021**, 126, 102094. [Google Scholar] [CrossRef] - Green, E.L.; Schroll, S. Brauer configuration algebras: A generalization of Brauer graph algebras. Bull. Sci. Math.
**2017**, 121, 539–572. [Google Scholar] [CrossRef] [Green Version] - Schroll, S. Brauer Graph Algebras. In Homological Methods, Representation Theory, and Cluster Algebras, CRM Short Courses; Assem, I., Trepode, S., Eds.; Springer: Cham, Switzerland, 2018; pp. 177–223. [Google Scholar]
- Cañadas, A.M.; Gaviria, I.D.M.; Vega, J.D.C. Relationships between the Chicken McNugget Problem, Mutations of Brauer Configuration Algebras and the Advanced Encryption Standard. Mathematics
**2021**, 9, 1937. [Google Scholar] [CrossRef] - Cañadas, A.M.; Espinosa, P.F.F.; Muñetón, N.A. Brauer configuration algebras defined by snake graphs and Kronecker modules. Electron. Res. Arch.
**2022**, 30, 3087–3110. [Google Scholar] [CrossRef] - Assem, I.; Skowronski, A.; Simson, D. Elements of the Representation Theory of Associative Algebras; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Andrews, G.E. The Theory of Partitions; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Sierra, A. The dimension of the center of a Brauer configuration algebra. J. Algebra
**2018**, 510, 289–318. [Google Scholar] [CrossRef] - Loday, J.L. Arithmetree. J. Algebra
**2002**, 258, 275–309. [Google Scholar] [CrossRef] [Green Version] - Boyvalenkov, P.; Dragnev, P.D.; Hardin, P.D.; Saff, E.B.; Stoyanova, M.M. Energy bounds for codes and designs in Hamming spaces. Des. Codes Cryptogr.
**2017**, 82, 411–433. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**This graph shows how we use topics described in the Background section to obtain the main results presented in this paper.

**Figure 2.**Snake graph ${\mathcal{G}}_{f}(2,2,2)=\mathcal{G}\left[5\right]$ and its perfect matchings.

**Figure 3.**Snake graph ${\mathcal{G}}_{f}(2,2,3)=\mathcal{G}[4,2]$ associated with a four-arrow string. The string module $M\left(w\right)$ over the corresponding Dynkin algebra of type $\mathbb{A}$ is obtained by replacing every vertex with a copy of a field $\mathbb{F}$. In such a case, arrows correspond to identity morphism.

**Figure 4.**Example of the Brauer quiver defined by the Brauer configuration ${\Gamma}^{2}$. Relations ${\alpha}_{1}^{1}{\beta}_{2}^{2}$, ${\alpha}_{1}^{2}{\beta}_{2}^{1}$, ${\alpha}_{1}^{1}{\alpha}_{1}^{2}{\alpha}_{1}^{1}$, ${\alpha}_{1}^{2}{\alpha}_{1}^{1}{\alpha}_{1}^{2}$, ${\beta}_{2}^{1}{\beta}_{2}^{2}{\beta}_{2}^{1}$, ${\beta}_{2}^{2}{\beta}_{2}^{1}{\beta}_{2}^{2}$, ${L}_{1}^{i}\sim {L}_{2}^{i}$ (where ${L}_{j}^{i}$ denotes the special cycle associated with the vertex j in polygon ${U}_{i}$, $i=1,2$) generate the admissible ideal ${I}_{{\Gamma}^{2}}$ for which the Brauer configuration algebra ${\Lambda}_{{\Gamma}^{2}}=\mathbb{F}{Q}_{{\Gamma}^{2}}/{I}_{{\Gamma}^{2}}$.

**Figure 5.**Indecomposable projective ${\Lambda}_{{\Gamma}^{2}}$-modules. Note that the number of composition series equals the number of non-truncated vertices in the corresponding polygon.

**Figure 8.**Examples of special snake graphs giving the minimal energy $\mathrm{min}\phantom{\rule{2.84544pt}{0ex}}E(1,{C}_{{\Gamma}^{j}},{h}_{\alpha})$, for $j=2$, $\alpha =1,2,3$.

**Figure 9.**Snake graphs associated with the energy values $\frac{191}{84}$, $\frac{2899}{504}$, $\frac{54,053}{30,240}$, and $\frac{894,547}{237,600}$.

**Table 1.**Valencies $val(i;{\Gamma}_{1}^{j})$ of vertices $i=1,2$, and 3 in ${\Gamma}_{1}^{j}$, $2\le j\le 6$.

$\mathbf{val}\left(1\right)$ | $\mathbf{val}\left(2\right)$ | $\mathbf{val}\left(3\right)$ | ${\mathbf{dim}}_{\mathbb{F}}\phantom{\rule{2.84544pt}{0ex}}{\mathit{\Lambda}}_{{\mathit{\Gamma}}^{\mathit{j}}}$ | |
---|---|---|---|---|

${\Gamma}_{1}^{2}$ | 2 | 2 | 8 | |

${\Gamma}_{1}^{3}$ | 6 | 4 | 5 | 72 |

${\Gamma}_{1}^{4}$ | 20 | 12 | 10 | 812 |

${\Gamma}_{1}^{5}$ | 70 | 40 | 30 | 9822 |

${\Gamma}_{1}^{6}$ | 252 | 140 | 100 | 124,112 |

$\mathit{j}\setminus \mathit{d}(\mathit{w},{\mathit{w}}^{\prime})$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|

2 | 1 | |||||||

3 | 6 | 4 | ||||||

4 | 25 | 34 | 32 | |||||

5 | 100 | 172 | 329 | 260 | ||||

6 | 390 | 754 | 1990 | 3126 | 2386 | |||

7 | 1516 | 3130 | 9983 | 21,638 | 32,481 | 23,058 | ||

8 | 5869 | 12,660 | 45,872 | 119,312 | 251,334 | 351,506 | 235,182 | |

9 | 22,746 | 50,570 | 202,205 | 589,306 | 1,519,120 | 3,001,666 | 3,944,860 | 2,486,618 |

**Table 3.**Energy values of codes ${C}_{{\Gamma}^{j}}$, $2\le j\le 9$ given by functions of the form ${h}_{\alpha}\left(t\right)\in \{1/i\mid 2\le i\le 9\}$, $1\le \alpha \le 3$.

$\mathit{\alpha}$ | $\mathit{j}\setminus {\mathit{h}}_{\mathit{\alpha}}\left(\mathit{t}\right)$ | 1/2 | 1/3 | 1/4 | 1/5 | 1/6 | 1/7 | 1/8 | 1/9 |
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 1/2 | |||||||

3 | 3 | 4/3 | |||||||

4 | 25/2 | 34/3 | 8 | ||||||

5 | 50 | 172/3 | 329/4 | 52 | |||||

6 | 195 | 754/3 | 995/2 | 3126/5 | 1193/3 | ||||

7 | 758 | 3130/3 | 9983/4 | 21,638/5 | 10,827/2 | 3294 | |||

8 | 5869/2 | 4220 | 11,468 | 119,312/5 | 41,889 | 351,506/7 | 117,591/4 | ||

9 | 11,373 | 50,570/3 | 202,205/4 | 589,306/5 | 759,560/3 | 3,001,666/7 | 986,215/2 | 2,486,618/9 | |

2 | 2 | 1/4 | |||||||

3 | 3/2 | 4/9 | |||||||

4 | 25/4 | 34/9 | 2 | ||||||

5 | 25 | 172/9 | 329/16 | 52/5 | |||||

6 | 195/2 | 754/9 | 995/8 | 3126/25 | 1193/18 | ||||

7 | 379 | 3130/9 | 9983/16 | 21,638/25 | 3609/4 | 3294/7 | |||

8 | 5869/4 | 4220/3 | 2867 | 119,312/25 | 13,963/2 | 351,506/49 | 117,591/32 | ||

9 | 211,373/2 | 50,570/9 | 202,205/16 | 589,306/25 | 379,780/9 | 3,001,666/49 | 986,215/16 | 2,486,618/81 | |

3 | 2 | 1/8 | |||||||

3 | 3/4 | 4/27 | |||||||

4 | 25/8 | 34/27 | 1/2 | ||||||

5 | 25/2 | 172/27 | 329/64 | 52/25 | |||||

6 | 195/4 | 754/27 | 995/32 | 3126/125 | 1193/108 | ||||

7 | 379/2 | 3130/27 | 9983/64 | 21,638/125 | 1203/8 | 3294/49 | |||

8 | 5869/8 | 4220/9 | 2867/4 | 119,312/125 | 13,963/12 | 351,506/343 | 117,591/256 | ||

9 | 11,373/4 | 50,570/27 | 202,205/64 | 589,306/125 | 189,890/27 | 3,001,666/343 | 986,215/128 | 2,486,618/729 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Moreno Cañadas, A.; Rios, G.B.; Serna, R.-J.
Snake Graphs Arising from Groves with an Application in Coding Theory. *Computation* **2022**, *10*, 124.
https://doi.org/10.3390/computation10070124

**AMA Style**

Moreno Cañadas A, Rios GB, Serna R-J.
Snake Graphs Arising from Groves with an Application in Coding Theory. *Computation*. 2022; 10(7):124.
https://doi.org/10.3390/computation10070124

**Chicago/Turabian Style**

Moreno Cañadas, Agustín, Gabriel Bravo Rios, and Robinson-Julian Serna.
2022. "Snake Graphs Arising from Groves with an Application in Coding Theory" *Computation* 10, no. 7: 124.
https://doi.org/10.3390/computation10070124