Exact dimer statistics and characteristic polynomials of cacti lattices

Summary The pruning method developed earlier by one of the authors (K.B.) combined with the operator method is shown to yield powerful recursive relations for generating functions for dimer statistics and characteristic polynomials of cacti graphs and cacti lattices. The method developed is applied to linear cacti, Bethe cacti of any length containing rings of any size, and cyclic cacti of any length and size. It is shown that exact dimer statistics can be done on any cactus lattice.Dedicated to Professor V. Krishnamurthy on the occasion of his 60th birthdayAlfred P. Sloan fellow; Camille and Henry Dreyfus teacher-scholar     

The use of Frame’s method for the characteristic polynomials of chemical graphs

The evaluation of characteristic polynomials of graphs of any size is an extremely tedious problem because of the combinatorial complexity involved in this problem. While particular elegant methods have been outlined for this problem, a general technique for any graph is usually tedious. We show in this paper that the Frame method for the characteristic polynomial of a matrix is extremely useful and can be applied to graphs containing large numbers of vertices. This method reduces the difficult problem of evaluating the characteristic polynomials to a rather simple problem of matrix products. The coefficients in the characteristic polynomial are generated as traces of matrices generated in a recursive product of two matrices. This method provides for an excellent and a very efficient algorithm for computer evaluation of characteristic polynomials of graphs containing a large number of vertices without having to expand the secular determinant of the matrix associated with the graph. The characteristic polynomials of a number of graphs including that of a square lattice containing 36 vertices are obtained for the first time.     

Properties and derivation of the fourth and sixth coefficients of the characteristic polynomial of molecular graphs—New graphical invariants

Some previously unknown relationships for determining the a ₄ and a ₆ coefficients of the characteristic polynomial for polycyclic aromatic hydrocarbons are presented for the first time. The structural information contained in these coefficients is more fully revealed. The equations derived for a ₄ and a ₆ allow one to determine the characteristic polynomial by inspection for many small molecular graphs. Some relationships for a ₈ and a ₁₀ are presented. The set of known graphical invariants (GI) or properties that remain unchanged in isomeric PAH6s is now shown to be GI={a ₄, a ₆+n ₀+2r ₆, a ₈ ^c , d _s+N_Ic, N_c, N_h, N_Ic+N_Pc, q, r}.Part VIII: A periodic table for polycyclic aromatic hydrocarbons     

Characteristic polynomials of chemical graphs via symmetric function theory

The characteristic polynomial corresponding to the adjacency matrix of a graph is constructed by using the traces of the powers of the adjacency matrix to calculate the coefficients of the characteristic polynomial via Newton's identities connecting the power sum symmetric functions and the elementary symmetric functions of the eigenvalues. It is shown that Frame's method, very recently employed by Balasubramanian, is nothing but symmetric functions and Newton's identities.     

Characteristic polynomials of linear polyacenes and their subspectrality

Coefficients of characteristic polynomials (CP) of linear polyacenes (LP) have been shown to be obtainable from Pascal’s triangle by using a graph factorisation and squaring technique. Strong subspectrality existing among the members of the linear polyacene series has been shown from the derivation of the CP’s. Thus it has been shown that the entire eigenspectrum of ann-ring LP is included in that of (2n + 1)-ring LP. Correspondence between the eigenspectra of linear chains and LP’s is brought out by a recently developed vertex-alternation and squaring algorithm.     

Subgraphs of pair vertices

Subgraphs obtained by applying several fragmentation criteria are investigated. Two well known criteria (Szeged and Cluj), and two new others are defined and characterized. An example is given for the discussed procedures. The matrix and polynomial representations of vertices composing each type of subgraphs were also given. Analytical formulas for the polynomials of several classes of graphs are derived. The newly introduced subgraphs/fragments, called MaxF and CMaxF, appear to have interesting properties, which are demonstrated.     

Characteristic polynomials of large graphs. On alternate form of characteristic polynomial [1]

A recursion exists between the absolute magnitudes of the coefficients of the characteristic polynomials of certain families of cyclic and acyclic graphs which makes their computation quite easy for very large graphs using a pencil-and-a-paper approach. Structural requirements are given for such families of graphs which are of interest to the problem of recognition defined in [1].     

Symmetry factoring of the characteristic equations of graphs corresponding to polyhedra

A systematic procedure is described which uses two-and three-fold symmetry elements in graphs to reduce their adjacency matrices to lead to corresponding factorings of their characteristic polynomials. A graph splitting algorithm based on this matrix reduction procedure is described. Applications of these methods to the factoring of the characteristic polynomials of 28 polyhedra with nine or less vertices are given. General expressions for the eigenvalues of prisms, pyramids, and bipyramids in terms of the eigenvalues of their basal or equatorial regular polygons are calculated by closely related matrix methods.     

Analysis of the topological dependency of the characteristic polynomial in its chebyshev expansion

The structural dependency (effect of branching and cyclisation) of an alternative form, the Chebyshev expansion, for the characteristic polynomial were investigated systematically. Closed forms of the Chebyshev expansion for an arbitrary star graph and a bicentric tree graph were obtained in terms of the “structure factor” expressed as the linear combination of the “step-down operator”. Several theorems were also derived for non-tree graphs. Usefulness and effectiveness of the Chebyshev expansion are illustrated with a number of examples. Relation with the topological index (Z _G ) was discussed. Operated for the U.S. Department of Energy by ISU under contract no. W-ENG-7405-82. Supported in part by the Office of Director     

Properties of “structure factor” of characteristic polynomial and a proof of Hosoya-Randić conjectures

Some properties of structure factor of characteristic polynomial are discussed and Hosoya-Randi conjectures are proved rigorously.     

Imminant polynomials of graphs

Summary The imminant polynomials of the adjacency matrices of graphs are defined. The imminant polynomials of several graphs [linear graphs (L _n ), cyclic graphs (C _n ) and complete graphs (K _n )] are obtained. It is shown that the characteristic polynomials and permanent polynomials become special cases of imminant polynomials. The connection between the Schur-functions and imminant polynomials is outlined.Cammile & Henry Dreyfus Teacher-scholar     

On self-reciprocal polynomials

In this paper we establish a sharp result concerning integral mean estimates for self-reciprocal polynomials.     

On Wiener‐type polynomials of thorn graphs

We derive the expressions of the ordinary, the vertex‐weighted and the doubly vertex‐weighted Wiener polynomials of a type of thorn graph, for which the number of pendant edges attached to any vertex of the underlying parent graph is a linear function of its degree. We also define variable vertex‐weighted Wiener polynomials and calculate them for the same type of thorn graphs. Copyright © 2009 John Wiley & Sons, Ltd.     

Discrimination and ordering of chemical structures by the number of walks

The use of the number of walks for the discrimination of graphs representing chemical structures is discussed. A highly selective graph-theoretical index based on the number of walks is defined. The index values were calculated for 661 acyclic and 376 cyclic structures. The selectivity of the new index is compared to that of some of the most selective previously defined indexes. The ordering of structures induced by the value of the index is also considered.Presented in part at the 7th International Conference on Computers in Chemical Research and Education, held in Garmisch-Partenkirchen, Germany, June 10–14, 1985     

On the special values of monic polynomials of hypergeometric type

Special values of monic polynomials y _n (s), with leading coefficients of unity, satisfying the equation of hypergeometric type

Acyclic systems with extremal Hückel π-electron energy

The main result of the present work is the proof that among acyclic polyenes C _n H _n+2, the linear isomer H₂C=(CH) _n–2=CH₂ has maximal HMO -electron energy. The 1,1-divinyl isomer (H₂C=CH)₂C(CH) _n–6=CH₂ has maximal -energy among branched acyclic systems. Among trees withn vertices, the star has minimal energy. A number of additional inequalities for HMO total -electron energy of acyclic conjugated systems are proved.     

Classification of chromatographic data using multidimensional polynomials

Summary A pattern recognition methodology has been developed for analysis of chromatographic data. The method uses a new class of multidimensional orthogonal polynomials developed by Cohen in conjunction with a supervised learning technique. The method is applicable to any chromatographic data for which classification into two or more categories is desired. The algorithm analyzes both elution times and peak areas. An application is shown for the analysis of organic acids in ascitic fluid obtained from patients with liver disorders. Classification of these patients for presence or absence of bacterial infection shows over ninety percent correct classification.