## 1 Introduction  
Theoretical models on complex networks have assumed a key role in numerous disciplines, ranging from computer science, physics, sociology, engineering, and medicine, to molecular, population biology, and deoxyribonucleic acid (DNA) sequences analysis [1, 2]. In the classical conception of DNA geometry, the double helix represents a ribbon constructed from smooth curves describing an idealized structure. How to read and recognize the primary structure of a DNA sequence seems to be a fundamental problem. In recent years, DNA sequencing and fragment assembly have received a much more specific attention for improving the reconstruction of full strands of DNA focused on the pieces of data to record. Fragment assembly due to imperfect data sets constitutes one of the most interesting challenges for researchers. In this paper, the concept of graph theory and fractal geometry are proposed a particular way of analyzing DNA sequences and their properties. Many scientific papers [3] dealing with DNA sequences analysis [4] have been published. However, the main aim of the work is to outline the advantage of applying graph theory and fractals in the study of lung cancer DNA sequences. This application allows the existence of hidden geometries to be shown, which can help underline more clearly the structure of a DNA sequence. An important consideration of this research is that this geometry has fractal structures. In order to achieve this goal, some basic steps must be observed, and they are indicated as follows: (1) Converting DNA nucleotides into a new coordinate system; (2) presenting DNA nucleotides as a path in a graph in the new coordinate system; (3) connecting points into a continuous graph; (4) using a method for estimating the Hurst exponent H; (5) using fractal geometry to determine the complexity of the graph; (6) constructing visibility graphs; and (7) calculating statistical and topological properties of the DNA network. In this perspective, this paper introduces an initial presentation of the concepts of graph theory, fractals, and pattern recognition for their possible use in the calculation of statistical and topological properties of the DNA network and the characterization of DNA sequences in lung cancer. The study of graph theory [5,6] was much elaborated in the twentieth century when the use of modeling techniques was implemented, and it has grown significantly to be routinely applied in various branches of science and engineering. Obviously, as special applications have multiplied in number and scope, the theory has developed considerably. It is known in mathematics that through graphs, complex relationships between many objects are commonly modelled. It is right to state that a graph is defined by a connection of nodes (also known as vertices) and edges. As an abstract data point, a node could represent anything, like nucleotides of DNA sequences. The application of mathematics to systems (such as a database of DNA sequences) with a large number of vertices and edges seems possible now as computers can be effectively adopted to provide solutions to such large graphs. Fractals [7, 8, 9, 10] are complex patterns occurring with self-similarly across different scales characterized by a fractal dimension. As mathematical descriptions of natural shapes, they become models generated by equations resulting in chaotic systems. In mathematical visualization, they are created from simple patterns that continue repeating over and over in a recursive process whose irregularity is very difficult to be described using classical geometry. Fractals can be thought of as images of dynamic systems. What makes fractal geometry more interesting for many scientists is its useful application in medicine, biology, and geology. In biology, chaotic systems can be used to show the rhythms of heartbeats, walking strides, and even the biological changes of aging. Fractals can be used to model the structures of nerve networks, circulatory systems, lungs, and even DNA [11].