# DNA graph characterization for the line digraph of dicycle with $\left\lfloor{\frac{n}{3}}\right \rfloor$ chords, $\infty$-digraph $C_n \cdot C_p$, and 3-blade-propeller $C_n \cdot C_p \cdot C_q$

## TL;DR

This paper demonstrates that certain line digraphs of specific cyclic and complex digraphs can be characterized as DNA graphs using $(
ame{alpha},k)$-labeling, expanding the class of DNA graphs relevant for DNA sequencing.

## Contribution

The paper introduces a construction of $(
ame{alpha},k)$-labeling to prove that line digraphs of dicycles with chords, $ty$-digraphs, and 3-blade-propellers are DNA graphs for various parameters.

## Key findings

- Line digraphs of dicycles with $loor{n/3}$ chords are DNA graphs for all $m extgreater 0$ and $n extgreater 5.
- Line digraphs of $ty$-digraphs $C_n imes C_p$ are DNA graphs for $n extgreater 2$ and certain $p$.
- Line digraphs of 3-blade-propellers $C_n imes C_p imes C_q$ are DNA graphs for $n extgreater 2$ and certain $p,q$.

## Abstract

DNA graph has important contribution in completing the computational step of DNA sequencing process. Using $(\alpha,k)$-labeling, several families of digraphs have characterized as DNA graphs. Dicycles and dipaths are DNA graphs, rooted trees and self adjoint digraphs are DNA graphs if and only if their maximum degree is not greater than four, while the $m^{\text{th}}$ line digraph of dicycle with one chord is a DNA graph for all $m\in\mathbb{Z}^+$. In this paper we construct $(\alpha,k)$-labeling to show that for all $m\in\mathbb{Z}^+$, the $m^{\text{th}}$ line digraph of dicycle $C_n$ with $\left\lfloor{\frac{n}{3}}\right \rfloor$ chords are DNA graphs for $n\geq 6$, and the $m^{\text{th}}$ line digraph of $\infty$-digraph $C_n\cdot C_p$ and 3-blade-propeller $C_n\cdot C_p \cdot C_q$ are DNA graphs for $n\geq 3$ and certain values of $p$ and $q$.