Matrices graph theory

Y. Yang, J. Lin, and C. Wang, Small regular graphs having the same path layer matrix, /. Graph Theory 39 (2002) 219-221. 

In graph theory, the conversion of the adjacency matrix into the distance matrix is known as the "all pairs shortest path problem",... 

Sink (in graph theory), 258 "Slack variables, 294 Slightly-ionized gases, 46 "Slow time, 362 Small parameter methods, 350 S-matrix, 599,649,692 Smirnova, T. S., 726 Smoluchowski, R., 745 Sokolov, A. V., 768 Sommerfeld, C. M., 722 Sonine polynomials, 25 Source (in graph theory), 258 Space group... 

In Chapter 1 we have stated that the classical structural theory is the only way to "visualise" the synthesis of a more or less complex organic compound. However, all or most of the information given by a structural formula can also be expressed.by a matrix (see also Appendix A-1). There are different kinds of matrices for example, the adjacency matrix J, which originates in graph theory and indicates only which atoms are bonded, or the connectivity matrix C, whose off-diagonal entries are the formal covalent bond orders. For instance, the corresponding matrices of hydrogen cyanide are ... 

Schultz, H. P., and E. B. Schultz, Topological Organic Chemistry 2. Graph Theory, Matrix Determinants and Eigenvalues, and Topological Indices of Alkanes. J. Chem. Inf. Comput. Sci., 1990 30, 27-29. 

Research Areas Linear Algebra, Matrix Theory, Numerical Analysis, Numerical Algebra, Geometry, Krein Spaces, Graph Theory, Mechanics, Inverse Problems, Mathematical Education, Applied Mathematics, Geometric Computing. 

The ideas of chemical graph theory have a special meaning for benzenoid hydrocarbons (BHs). As the structures of BHs do not allow for the cis/trans isomers, there is a rigorous correspondence between the adjacency matrices of BHs and their properties. In other words, any property JP(G) of the benzenoid molecule G is a function of only its adjacency matrix A(G)... 

Much of the material we present in this paragraph is discussed exhaustively in two classical textbooks on the chemical graph theory [7, 8] and therefore we review it only very briefly. Let A(G) be the adjacency matrix pertinent to a molecular graph... 

From vertex to vertex. In the graph theory the matrix elements Aij can be interpreted as follows an Atj is the number of unitary walks between the vertices i and /. Then the product of two elements of the matrix A, ArjAj is equal to 1 if the vertex r is connected with the vertex /, and the latter in its turn is connected with s, i.e. between r and s there is a walk of length 2 passing through /. If there is no such a walk, Aj.jAjs = 0. 

Topological orbitals... Probably the smartest of our readers have already appreciated the close relationship between the graph theory and the Hiickel method. Actually in quantum chemistry there is a great variety of problems in which the Hamiltonian of a molecule can be written in a matrix form as a one-valued function of the topological matrix of that molecule ... 

The Mlowing transformation of a topological matrix of a graph plays an important part in the graph theory ... 

The correspondence between mathematical graph theory and classical chemical structure theory is manifested in Table I. A widely used mathematical representation for graphs is the adjacency matrix (A). The rank of this matrix equals the number of the vertices (atoms), and its entries a,y are equal to either 0 or 1 ... 

Barysz, M., Jashari, G., Lall, R.S., Srivastava, A.K. and Trinajstic, N. (1983a). On the Distance Matrix of Molecules Containing Heteroatoms. In Chemical Applications of Topology and Graph Theory (King, R.B., ed.), Elsevier, Amsterdam (The Netherlands), pp. 222-230. 

Rouvray, D.H. (1976). The Topological Matrix in Quantum Chemistry. In Chemical Applications of Graph Theory (Balaban, A.T., ed.), Academic Press, New York (NY), pp. 175-222. 

Schultz, H.P., Schultz, E.B. and Schultz, T.P. (1992). Topological Organic Chemistry. 4. Graph Theory, Matrix Permanents, and Topological Indices of Alkanes. J.Chem.lnf.Comput.ScL, 32, 69-72. 

Schultz, H.P. and Schultz, T.P. (1992) Topological organic chemistry. 5. Graph theory, matrix hafnians and pfafflans, and topological indices of alkanes. /. Chem. Inf. Comput. Sci., 32, 364—368. 

The fundamental fact which makes it possible to apply Graph Theory in quantum chemistry is that the molecular Hamiltonian for a wide class of compounds can be written in the matrix form as a unique function of the graph corresponding to the molecule under consideration, i.e. 

Graph Theory and Molecular Orbitals From elementary matrix algebra it is known that... 

Traditional notation is introduced here. A pattern of the bonds in the system is obtained when all p s are set equal to unity and the oc s are omitted. The remaining matrix is called the topological matrix and has been the subject of studies in graph theory. 

In the spirit of chemical graph theory, a numerical expression was developed to encode the structure information represented in the paths. The numerical value for the path from vertex i to vertex j is the entry f,y in a topological state matrix T. The topological state value, T, for vertex i is, then, the sum of the entries for row i. 

---