Analytical eigenspectra for the graphs of linear chains and cycles with alternant edge weights has been derived use two independent methods, namely, characteristic polynomial graph squaring. In former method rotational symmetry trigonometric identity have exploited. These methods along expressions so obtained found to be very useful in expressing analytical eigensolutions some important as well novel benzenoids, example, p-methylene poly(p-phenylene), cylindrical zigzag graphene, carbon...

A graph of {X, Y}-cyclopolyacene with n hexagonal rings has been presented that contains four orbits, which orbits 1 and 4 are occupied by the X-type vertex 2 Y-type, or vice versa. Eigensolutions for such a have derived in analytical form through use rotational symmetry followed plane symmetry. Varying X ( = C, N, B, …) Y several types cyclopolyacene may be obtained. Eigenvalue-expressions systems containing N B shown their total π-electron energies 2–6 calculated help expressions developed.

Abstract The compounds of poly (p-phenylene) are important materials for electro-optical and electronic applications. Algorithms the calculation distances hence Wiener indices (W) linear cylindrical systems have been developed that require only number hexagonal rings present in systems. Ambient conditions density bulk modulus (p-phenylenes) with two to six phenyl correlated W 1/4. Cylindrical 5 6 found occur as intervening strands between caps fullerene cages. Keywords:...

Abstract Three classes of pendent vertex- and pendant edge-weighted graphs linear chains (class I), stars II), cycles III) have been presented. These (particularly class I represent heteroconjugated π-systems. Sometimes such appear as factored subgraphs some complicated graphs. The eigenspectra these found out in analytical forms. utilized i) to calculate the band (i.e., HOMO–LUMO) gaps graphs, ii) find three inversely proportional (with pairs eigenvalues), iii) express forms along with...

A graph theoretical procedure for solving multistep coupled kinetic rate equations and thereby obtaining the concentrations of species involved in reaction has been developed. The method so developed illustrated with some well-known schemes.

Graph theoretical solutions for kinetic rate equations of some reaction networks involving linear chains and cycles have been derived; condensation polymerization long chain radioactive decay come under the purview former whereas interconversion species in later. The reactions considered here proceed monotonically to steady states with time cycle all irreversible steps has found either periodic or monotonic evaluation concentrations depending on values constants involved paths. In case a...

A general approach to determine the matching polynomial (MP) of a graph with two parts connected by an edge is presented in matrix product that ultimately used deducing recursion formulas for obtaining MP coefficients linear and cylindrical poly(p-phenylene) (PPP) graphs. The Hosoya indices PPPs are derived terms immediately preceding graphs as well analytical forms use transfer matrices. Ambient condition density bulk modulus 2–6 phenyl rings have been found correlate logarithm their...

An algorithm has been developed for the construction of planar graphs two series IPR fullerenes general formula C50 + 10n and C60 12n (n = 1, 2, 3,…) maintaining 5- 6-fold rotational symmetry, respectively. Some important utilities these have shown. Simple rules then factorising in order to obtain eigenvalues corresponding totally symmetric irreducible representations C5 C6 subgroups full point groups fullerenes.

Three classes of reciprocal graphs, viz. monocycle (GCn), linear chain (GLn) and star (GKn) with pairs eigenvalues (λ, 1/λ), are well known. Reciprocal graphs (GCn) obtained by putting a pendant vertex to each simple (Cn) (Ln), respectively. A graph such kind is attaching the central (n − 1) peripheral vertices (K1, (n−1)). An n-fold rotational axis symmetry for GCn 1)-fold GKn have been exploited obtaining their respective condensed graphs. The GLn has generated from that incorporating...

Abstract A graph theoretical procedure for obtaining eigenvalues of linear chains and cycles having alternant vertex weights ( h 1 , 2 …) the same edge weight k ) have been developed. The some complicated graphs, such as graphs polyacenes, methylene‐substituted polyacenes cylindrical polyacene strips, stack reciprocal shown to be generated in closed analytical forms by this procedure. Many represent chemically important molecules or radicals. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem,

Abstract Two types of topological indices, Wiener and Hosoya three classes reciprocal graphs (namely , ) have been shown to be expressed in terms the number pendant vertices (n). The index is analytical form whereas either by recurrence relations or directly matrix products for which a facile computer program can written. It has also that obtained form. Acknowledgement Authors are thankful University Grants Commission, New Delhi, granting financial assistance extended through DSA project...

Abstract Cardinalities of three classes the recently investigated reciprocal graphs, viz. L n + ( p ), C and K 1, −1 ) have been shown to be given by two recurrence relations for first types an analytical formula third type. Application these calculate bond orders some conjugated molecules has shown. In case both topological (TBO) Hückel remain nearly independent . This explained deriving expressions TBOs in form continued fractions. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem,

Abstract Characteristic polynomial coefficients of three classes graph, namely Ln + n(p), Cn n(p) and K 1,n-1 which are known to have reciprocal pairs eigenvalues, been shown be generated by simple manipulation the Pascal's triangle.

Abstract A method for construction of the characteristic polynomial (CP) coefficients three classes reciprocal graphs, viz., L n + ( p ), C and K 1, −1 has been developed that requires only value n. The working formulas have expressed in matrix product form, computer programs which can easily be developed. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Three types of graphs linear chains, viz. chains with unit increment or decrement in weight on one terminal vertex, both the vertices and vertex that other have been considered. The symmetry plane fragmentation graph squaring techniques exploited to express eigenspectra such analytical form, subsequently used cycles alternant weights. derived expressions for obtain polyacenes, methylene-substituted polyacenes cylindrical polyacene strips form.

Algorithm for obtaining characteristic polynomial (CP) coefficients of an alternant edge-weighted cycle is used to arrive at the algorithm that cycloparaphenylene (CPP) graphs in matrix product form. The gives a recursive relation expressing sum CP CPP terms its two immediately preceding analogues which ultimately ends up with use transfer analytical coefficients, being combinatorial nature, found be as topological index showing much similarity Hosoya (sum all matching coefficients),...

Graphs of linear chains and/or cycles with homogeneous alternant vertex weights have been considered. If one puts weight(s) on the terminal (vertices) or edge(s) such chains...

Recurrence relation for the cardinalities of linear and cylindrical poly(p-phenylene) (PPP) compounds has been developed that requires two their immediate lower homologues. Such recurrence reduces into analytical expressions under transfer matrix formalism. Ambient condition density bulk modulus PPPs are found to bear excellent correlation with inverse logarithm cardinalities. Topological bond orders obtained from such have good correlations respective Hückel orders.

Abstract Threefold rotational symmetry has been used to develop an algorithm for the construction of planar graphs IPR fullerenes and factorize their characteristic polynomials. Two series formula C 60+12 n 60+18 have thus obtained. The shown be useful predicting nature variation point groups with increased , counting number 13 nuclear magnetic resonance (NMR) signals (along relative intensities), also obtaining a large part eigenspectra. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem,

Formulas for the characteristic polynomial (CP) coefficients of three classes (n + p)-vertex graphs, i.e. linear chains, cycles and stars where p pendant vertices are attached to n base in one-to-one correspondence (p = 0, 1, 2, …, n), have been developed. Such graphs become reciprocal chains if n. The n-vertex star follow same rule as paths cycles, they on adding a vertex each vertices. formulas so developed expressed matrix product analytical forms that require only values calculation...

Eigensolutions of {X( = C,B,N),Y( C,B,N)}-cyclacene graphs with next nearest neighbor (nnn) interactions have been obtained in analytical forms by adapting n-fold rotational symmetry followed two-fold (or a plane symmetry). Expressions eigensolution indicate the subspectral relationship among such cyclacenes an even number hexagonal rings e.g., eigenvalues {X,Y}-di-cyclacene are found eigenspectra all cyclacenes. Total π-electron energies and highest occupied molecular orbital lowest...