# Different DNA Sequencing Using DNA Graphs: A Study
## 2 DNA Library  
A DNA library may be thought of as a collection of DNA sequences. A DNA library, like a book or any other type of data library, may be used to store and distribute information. A DNA chain serves as the data carrier in a similar type of library. Typically, these chains reflect natural genetic data derived from biochemical investigations. DNA can provide information about gene expression events or aid in the detection of genetic illnesses. However, some applications necessitate DNA chains with special characteristics that are seldom seen in natural DNA molecules. DNA computing, which was developed in 1994 by [PERSON] [1], is an example of such an application. Graph vertices are encoded as randomly produced DNA chains of length \(l\), while arcs are encoded as a concatenation of two chains: one supplementary to the last \(1/2\) nucleotides of the preceding vertex and one additional to the first \(1/2\) nucleotides of the succeeding vertex, the Hamiltonian path problem was solved. In a perfect scenario, vertices and arcs would hybridize; nevertheless, random chain creation might result in hybridization between arcs and vertices non-existent in that graph. The research provides an approach for building DNA libraries composed of chains with a low proclivity for hybridization. As a result, this kind of library may later be utilized for encoding a collection of vertices. This approach is significant from a mathematical standpoint. As a result, a generalized de Bruijn sequence [2] lacks complementarity. The algorithm's first draught, on the basis of de Bruijn graphs, was published in [8], and it was modified further in [9]. The approach relies on graph theory, more precisely on lexical graphs, which are de Bruijn graphs that have been expanded. In [10], lexical graphs were introduced, and independently similar ideas were found in [11,12]. In [13], they were named word graphs; in [12], they were referred to as alphabet overlap digraphs. Each Eulerian cycle in a de Bruijn graph correlates to a de Bruijn sequence [2]. Every Eulerian cycle in a lexical graph corresponds to a lexical sequence, which is a de Bruijn sequence that has been expanded. De Bruijn sequences with several shifts were presented in [9] and independently in [14], where they are referred to as lexical sequences. Ref. [14] developed de Bruijn sequences with multiple shifts to solve the Frobenius problem in a free monoid [15]. Labeled graphs are subgraphs of de Bruijn graphs that relate to the DNA and labeled graphs described in [4]. Induced subgraphs of lexical graphs are analogically base labeled graphs [16]. DNA graphs are employed in DNA sequencing, for example [4,9] (see Figure 2). All types of graphs discussed above have the attribute of having labels given to their vertices [12,15]. These labels are words with a fixed measurement that is determined by an alphabet. Any DNA chain may be regarded as a sequence using the letters A, C, T, and G. The DNA chains being directed, they contain two separate endpoints, one designated \(3^{\prime}\) and the other \(5^{\prime}\) (for example, \(5^{\prime}\)-ACTG-\(3^{\prime}\) or, equivalently, \(3^{\prime}\)-GTCA-\(5^{\prime}\)). Because the direction of the DNA chain is critical for the algorithm, we shall assume one fixed orientation, such as \(5^{\prime}\) to \(3^{\prime}\) (thus, ACTG will always imply \(5^{\prime}\)-ACTG-\(3^{\prime}\)). In DNA, hybridization happens only between chains that run in reverse directions; hence, chain AATCCG is complementary to chain CGGATT. The main aim of this study is to analyze the visualization of DNA sequencing using graph theoretic approaches.