# Novel Graphical Representation and Numerical Characterization of DNA Sequences
## 1 Introduction  
The rapid development of DNA sequencing techniques has resulted in explosive growth in the number of DNA primary sequences, and the analysis and comparison of biological sequences has become a topic of considerable interest in Computational Biology and Bioinformatics. The traditional measure for similarity analysis of DNA sequences is based on multiple sequence alignment, which uses dynamic programming techniques to identify the globally optimal alignment solution. However, the sequence alignment problem is NP-hard (non-deterministic polynomial-time hard), making it infeasible for dealing with large datasets [1]. To overcome the limitation, a lot of alignment-free approaches for sequence comparison have been proposed.  
The basic idea behind most alignment-free methods is to characterize DNA by certain mathematical models derived for DNA sequence, rather than by a direct comparison of DNA sequences themselves. Graphical representation is deemed to be a simple and powerful tool for the visualization and analysis of bio-sequences. The earliest attempts at the graphical representation of DNA sequences were made by [PERSON] and [PERSON] in 1983 [2]. Afterwards, a number of graphical representations were well developed by researchers. For instance, by assigning four directions defined by the positive/negative \(x\) and \(y\) coordinate axes to the four nucleic acid bases, [PERSON] [3], [PERSON] [4, 5], and [PERSON] [6] introduced three different 2-D graphical representations, respectively. While [PERSON] [7] proposed a chaos game representation (CGR) of DNA sequences, in which the four corners of a selected square are associated with the four bases respectively. In 2000, [PERSON] _et al._[8] generalized these 2-D graphical representations to a 3-D graphical representation, in which the center of a cube is chosen as the origin of the Cartesian (\(x\),\(y\),\(z\)) coordinate system, and the four corners withcoordinates (+1,-1,-1), (-1,+1,-1), (-1,-1,+1), and (+1,+1,+1) are assigned to the four bases. Some other graphical representations of bio-sequences and their applications in the field of biological science and technology can be found in [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24].  
Numerical characterization is another useful tool for sequence comparison. One way to arrive at the numerical characterization of a DNA sequence is to associate the sequence with a vector whose components are related to \(k\)-words, including the single nucleotide, dinucleotide, trinucleotide, and so on [25; 26; 27; 28; 29; 30]. In addition, the numerical characterization can be accomplished by associating with a graphical representation given by a curve in the space (or a plane) structural matrices, such as the Euclidean-distance matrix (ED), the graph theoretical distance matrix (GD), the quotient matrix (D/D, M/M, L/L), and their \"higher order\" matrices [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. Once a matrix representation of a DNA sequence is given, some matrix invariants, e.g. the leading eigenvalues, can be used as descriptors of the sequence. This technique has been widely used in the field of biological science and medicine, and different types of matrices are defined to construct various invariants of DNA sequences. However, the order of these matrices is equal to \(n\), the length of the DNA sequence considered. A problem we must face is that the calculation of these matrix invariants will become more and more difficult with larger \(n\) values [17; 24; 32].  
In this paper, based on all of the 2-combinations of the multiset \(\{\infty\cdot\text{A},\infty\cdot\text{G},\infty\cdot\text{C},\infty\cdot\text{T}\}\), we propose a novel graphical representation of DNA sequences. Then, according to the idea of \"piecewise function\", we describe a particular scheme that transforms the graphical representation of DNA into a cell-based descriptor vector. The introduced vector leads to more simple characterizations and comparisons of DNA sequences.