## 1 Introduction  
In the task of comparison and analysis of biological sequences, choosing a type of DNA/protein representation is an important step. The usual representation of the primary structure of DNA is a string of four letters: A (adenine); G (guanine); C (cytosine); and T (thymine). This expression is called a letter sequence representation (LSR) or a DNA primary sequence. Similarly, a protein primary sequence is usually expressed in terms of a series of 20 letters, which denote 20 different amino acids. The sequence encodes information of the corresponding structure and function in a living organism. However, it is difficult to obtain the information from the representation of a primary sequence directly. Therefore, various sequence representation techniques have been developed for encoding bio-sequences and extracting the hidden information. Graphical representation of DNA is a useful tool for visualizing and analyzing DNA sequences. By using the tool, one can obtain a route to condense the information coded by DNA primary sequences into a set of invariants [1,2]. Early attempts towards graphical representations of DNA were made by [PERSON] and [PERSON] in 1983 [3], [PERSON] in 1985 [4], and [PERSON] in 1985 [5]. Afterwards, more graphical representations of DNA sequences were well developed by researchers [1,2,6,7,8,9,10,11,12,13,14,15]. In comparison with DNA, graphical representations of proteins emerged only very recently [2,16,17,18,19,20,21,22,23,24,25,26,27]. As a matter of fact, most of the graphical representations of DNA involve some degree of arbitrariness, such as the selection of directions to be assigned to individual bases. For a string like DNA sequence over an alphabet with size 4, there are \(4!=24\) possible ways of assigning 4 directions to 4 nucleic acid bases. If these methods are directly extended to protein sequences, the corresponding figure is \(20!\approx 2.433\times 10^{18}\). It is impracticable to represent one protein sequence by such an enormous number of graphs. This is probably the most important reason why protein graphical representations have not been advanced [19; 23]. It is found that reducing the alphabet or fixing the directions assigned to amino acid residues plays an important role in addressing this problem. For details, we refer to some recent publications [2; 16; 21; 23; 24; 28]. Matrix representation of a biological sequence is another powerful tool for characterization and comparison of sequences. These matrices include: The frequency matrix; Euclidean-distance matrix (_ED_); graph theoretical distance matrix (_GD_); line distance matrix (_LD_); quotient matrix (_D/D_, _M/M_, _L/L_); and their \"higher order\" matrices [1; 2; 12; 13; 20; 21; 27; 29; 30]. Among them, _ED_, _GD_, _L/L_, etc., are derived from a graphical representation. For example, _L/L_ is a symmetric matrix whose diagonal entries are zero, while other entries are defined as the quotient of the Euclidean distance between two points of the graph and the sum of geometrical lengths of edges between the two points. Once the matrix is given, some of matrix invariants can be used as descriptors of the sequence. Eigenvalues of a matrix are one of the best-known matrix invariants [31]. In fact, two graphs are isomorphic if and only if their adjacency matrices are similar. It is of interest to note that similar matrices have the same eigenvalues. Among all the eigenvalues, the leading eigenvalue often plays a special role and has been widely used in the field of biological science and chemistry. However, a problem we must face is that the calculation of the eigenvalue will become more and more difficult with the order of the matrix large. ALE-index is an alternative invariant we proposed in 2005 [32]. The ALE-index can be viewed as an Approximation of the Leading Eigenvalue (ALE) of the corresponding matrix (it is just in this sense that it is called 'ALE'-index), while it is much simpler for calculation than the latter.