Xiaofeng Guo

On the characterization of DNA primary sequences by triplet of nucleic acid bases

We consider construction of a set of smaller 4 x 4 matrices to represent DNA primary sequences which are based on enumeration of all 64 triplets of nucleic acids bases. The leading eigenvalue from the constructed matrices has been selected as an invariant for construction of a vector to characterize DNA. Additional invariants considered of the derived condensed matrices of DNA include a 64-component vector, the components of which consist of ordered triplets XYZ, with X, Y, Z = A, C, G, T. Construction of similarity/dissimilarity tables based on different invariants for a set of sequences of DNA belonging to the first exon of the beta-globin gene of eight species illustrates the utility of newly formulated invariants for DNA.

A novel 2-D graphical representation of DNA sequences of low degeneracy

Some 2-D and 3-D graphical representations of DNA sequences have been given by Nandy, Leong and Mogenthaler, and Randic et al., which give visual characterizations of DNA sequences. In this Letter, we introduce a novel graphical representation of DNA sequences by taking four special vectors in 2-D space to represent the four nucleic acid bases in DNA sequences, so that a DNA sequence is denoted on a plane by a successive vector sequence, which is also a directed walk on the plane. It is showed that the novel graphical representation of DNA sequences has lower degeneracy and less overlapping.

Wiener matrix: source of novel graph invariants

We report some properties of new graph matrices which appear to offer novel graph invariants of potential interest in structureproperty studies. The matrices are constructed by generalizing Wiener’s procedure for evaluation of Wiener numbers in alkanes. Among the invariants considered we particularly examined the sequences generated by summing the entries in the matrix for vertices at the same distance from one another. These numbers may be viewed as “higher” Wiener numbers in analogy with “higher“ connectivity indices. We have listed the higher Wiener numbers of alkanes up to n = 9 carbon atoms and also report several recursions for the construction of these invariants for selected families of acyclic graphs. Briefly, we have outlined how the Wiener matrix can be extended to cyclic systems, while in the concluding comments we have outlined an extension of the Wiener matrix to molecules having heteroatoms. The significance of the matrices as a source of graph invariants is precisely in this possibility to go beyond simple models of molecular graphs and extend graph invariants of interest to molecules having different kinds of atoms.

Z -transformation graphs of perfect matchings of hexagonal systems

Abstract Let H be a hexagonal system. We define the Z -transformation graph Z( H ) to be the graph where the vertices are the perfect matchings of H and where two perfect matchings are joined by an edge provided their symmetric difference is a hexagon of H . We prove that Z ( H ) is a connected bipartite graph if H has at least one perfect matching. Furthermore, Z ( H ) is either an elementary chain or graph with girth 4; and Z ( H ) - V m is 2-connected, where V m is the set of monovalency vertices in Z ( H ). Finally, we give those hexagonal systems whose Z -transformation graphs are not 2-connected.

Boundary uniqueness of fuseness

It is shown that a geometrically planar fusene is uniquely determined by its boundary edge code. Surprisingly, the same conclusion is not true in general but holds for geometrically planar and non-planar fusenes with at most 25 hexagons, except for two particular cases. In addition, it is proved that two fusenes with the same boundary edge code have the same number of hexagons.

k-coverable coronoid systems

A coronoid systemG isk-coverable if for everyk (or fewer) pairwise disjoint hexagons the subgraph, obtained fromG by deleting all thesek hexagons together with their incident edges, has at least one perfect matching. In this paper, some criteria are given to determine whether or not a given coronoid system isk-coverable.

Mathematical properties and structures of sets of sextet patterns of generalized polyhexes

Several definitions of sextet patterns and super sextets of (generalized) polyhexes have been given, first by He Wenjie and He Wenchen [1], later by Zhang Fuji and Guo Xiaofeng [2], and by Ohkami [3], respectively. The one-to-one correspondence between Kekulé and sextet patterns has also been proved by the above authors using different methods. However, in a rigorous sense, their definitions of sextet patterns and super sextets are only some procedures for finding sextet patterns and super sextets, not explicit definitions. In this paper, we give for the first time such an explicit definition from properties of generalized polyhexes, and give a new proof of the Ohkami-Hosoya conjecture using the new definition. Furthermore, we investigate mathematical properties and structures of sets of generalized polyhexes, and prove that thes-sextet rotation graphRs(G) of the set of sextet patterns of a generalized polyhexG is a directed tree with a unique root corresponding to theg-root sextet pattern ofG.

Enumeration of helicenes

Abstract Helicenes are simply connected, geometrically non-planar polyhexes. Enumeration results for these systems are reported. The C n H s formulae for helicenes are treated theoretically, and some of the C n H s helicene isomers are enumerated. The total numbers of polyhexes with eight and nine hexagons are corrected.

Recursive method for enumeration of linearly independent and minimal conjugated circuits of benzenoid hydrocarbons

The linearly independent and minimal conjugated (LM-conjugated) circuits of benzenoid hydrocarbons (BHs) play the central role in the conjugated circuit model. For a general case, the enumeration of LM-conjugated circuits of BHs may be tedious as it requires construction of all Kekule structures. In the present paper, we investigate the properties and the construction of minimal conjugated circuits of BHs, and give the necessary and sufficient condition for a set of conjugated circuits of a BH to be linearly independent and minimal. Furthermore, we establish some recursive relations for enumeration of LM-conjugated circuits for several classes of BHs, one class of which consists of all the BHs containing no crown subunit. By these recursive relations, the summation expressions of LM-conjugated circuits (LMC-expressions) of the several classes of BHs can be directly obtained from the LMC-expressions and the Kekule structure counts of their subgraphs.

k -cycle resonant graphs

Abstract A connected graph G is said to be k -cycle resonant if, for 1 ⩽ t ⩽ k , any t disjoint cycles in G are mutually resonant, that is, there is a perfect matching M of G such that each of the t cycles is an M -alternating cycle. In this paper, we at the first time introduce the concept of k -cycle resonant graphs, and investigate some properties of k -cycle resonant graphs. Some simple necessary and sufficient conditions for a graph to be k -cycle resonant are given. The construction of k -cycle resonant hexagonal systems are also characterized.