# A New Class of Graph Grammars and Modelling of Certain Biological Structures

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## Abstract

**:**

## 1. Introduction

## 2. Graph Grammars

#### Node Replacement Graph Grammars

**Definition**

**1**

- Σ is an alphabet used to label nodes,
- Γ is a collection of terminal symbols in Σ,
- A production rule in P, $p:A\to D$ acts on the mother node labelled A,
- C is a collection of embedding instructions in $\Sigma \times \Sigma $,
- ${G}_{S}$ is the initial graph.

**Definition**

**2**

- Σ and Γ are sets of symbols used to label nodes and edges respectively,
- Δ and Ω are the collections of terminal symbols in Σ and Γ respectively,
- A production rule in P, $p:A\to (D,C)$ acting on the mother node M with label A has a collection C of connection instructions $(a,p\mid q,B)$ associated with it. Here B is a node in D and x with label a is one of the neighbors of M. The edge p which connected x and M is removed and an edge q is established between x and B.
- ${G}_{S}$ is the initial graph.

## 3. Non-Confluent Edge and Node Controlled Embedding $(\mathit{nc}$-$\mathit{eNCE})$ Graph Grammar

**Definition**

**3**

- Σ and Γ are sets of symbols used to label nodes and edges respectively,
- Δ and Ω are the collections of terminal symbols in Σ and Γ respectively,
- A production rule in P, $p:A\to (D,C)$ acting on the mother node M with label A has a collection C of connection instructions $(a,p\mid q,B)$ associated with it. Here x with label a is a neighbor of M and B is a node in D. The edge p which connected x and M is removed and a new edge q is established between x and B.
- ${G}_{S}$ is the initial graph,
- The regular control, $R\left(P\right)$, regulates the sequence of application of the production rules.

#### 3.1. nc-$eNCE$ Graph Grammar with Deletion $(dnc$-$eNCE)$

**Definition**

**4.**

- Σ is an alphabet used to label nodes,
- $\Delta \subset \Sigma $ is a collection of terminal symbols,
- Γ is an edge labelling alphabet,
- Ω is the edge labels of the final graph, [22].
- P contains in addition to productions defined in Definition 2, the rules $A\to \epsilon $ where, M labelled A is the node to be deleted. The connection instruction for this rule has the format $\left(\right(a,x),z,(y,b\left)\right)$. In this rule the edges labelled x, y connecting M to a and b respectively are removed, and a new edge with label z is established between the two nodes.
- ${G}_{S}$ is the initial graph.
- The regular control $R\left(P\right)$ regulates the sequence of application production rules.

**Example**

**1.**

#### 3.2. Non-Confluent $eNCE$ Graph Grammar with $\psi $ Labelled Edges $(\psi nc$-$eNCE)$

**Definition**

**5.**

- Σ is an alphabet used to label nodes,
- $\Delta \subset \Sigma $ is a collection of terminal symbols,
- $\Gamma \cup \psi $ is the edge labelling alphabet,
- Ω is the collection of edge labels of the final graph,
- P contains rules similar to those in Definition 2. In addition we introduce a special edge label ψ which acts as follows: While concatenating two graphs together or embedding a daughter graph in a mother graph, this edge can be introduced between two nodes with terminal labels. The connection instruction associated with this edge has the format $(x,\psi ,y)$. This label can be bypassed or ignored while specifying the graph language.
- ${G}_{S}$ is the initial graph.
- The regular control $R\left(P\right)$ regulates the sequence of application of the production rules.

**Example**

**2.**

## 4. Generation of Certain Graph Classes

#### 4.1. Wheel Graphs

**Definition**

**6**

**Lemma**

**1.**

**Proof.**

#### 4.2. Complete Bipartite Graphs

**Definition**

**7**

**Lemma**

**2.**

**Proof.**

#### 4.3. Binary Tree

**Definition**

**8**

**Lemma**

**3.**

**Proof.**

## 5. Generative Power of $\mathit{nc}$-$\mathit{eNCE}$ Graph Grammars

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 6. Modelling of Biological Structures

#### 6.1. Modelling of Parallel $\beta $-Sheet Structures

#### 6.2. Modelling of Anti-Parallel $\beta $-Sheet Structures with a Semi-Greek Key Conformation

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Vijayakumar, J.; Mathew, L.; Nagar, A.K.
A New Class of Graph Grammars and Modelling of Certain Biological Structures. *Symmetry* **2023**, *15*, 349.
https://doi.org/10.3390/sym15020349

**AMA Style**

Vijayakumar J, Mathew L, Nagar AK.
A New Class of Graph Grammars and Modelling of Certain Biological Structures. *Symmetry*. 2023; 15(2):349.
https://doi.org/10.3390/sym15020349

**Chicago/Turabian Style**

Vijayakumar, Jayakrishna, Lisa Mathew, and Atulya K. Nagar.
2023. "A New Class of Graph Grammars and Modelling of Certain Biological Structures" *Symmetry* 15, no. 2: 349.
https://doi.org/10.3390/sym15020349