The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

Electron-pair radial density functions

We introduce and discuss a generalized electron-pair radial density function G(q; a) that represents the probability density for the electron-pair radius |r ₁+ar ₂| to be q, where a is a real-valued parameter. The density function G(q; a) is a projection of the two-electron radial density D ₂(r ₁, r ₂) along lines r ₁ + ar ₂ ± q = 0 in the r ₁ r ₂ plane onto a point in the qa plane, and connects three densities S(s), D(r), and T(t), defined independently in the literature, as a smooth function of a: For an N-electron (N ≥ 2) system, S(s) = G(s; + 1), D(r) = 2G(r; 0)/(N − 1), and T(t) = G(|t|;−1)/2, where S(s) and T(t) are the electron-pair radial sum and difference densities, respectively, and D(r) is the single-electron radial density. Simple illustrations are given for the helium atom in the ground 1s² and the first excited 1s2s ³S states.     

Vibrational spectra of icosahedral (Ih and I) fullerenes

On the bases of the topological structures of the three big classes of icosahedral fullerenes: (1) C_n(I_h, n=60h²; h=1, 2,…), (2) C_n(I_h, n=20h²; h=1, 2,…), and (3) C_n(I, n=20(h²+hk+k²), h>k; h, k=1, 2,…), we derived formulas for the decomposition of their nuclear motions into irreducible representations. Hence, we obtained the infrared and Raman active modes for all of the icosahedral (I_h and I) fullerenes theoretically. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 113–117, 1998     

Some properties of the structure factor S(q) in two-dimensional classical liquids near freezing

We explore general properties of the main peak of the structure factor S(q) near the melting temperature T _melt in liquids confined in two dimensions, especially for the one component plasma model and for monatomic liquids interacting through inverse twelfth-power potentials. Those properties are the height of the peak, S(q _m), where q _m is the position of maximum in the peak, and the ratio between S(q _m) and q _m/Δq, where 2Δq is the width of the peak. The results obtained are then compared with those for similar systems in three dimensions. Other magnitude that we use to compare two-dimensional and three-dimensional simple liquids is r _m/Δr, where r _m is the position of the main peak in the pair distribution function g(r) and 2Δr is the width of that peak.     

New upper bounds on Zagreb indices

The first Zagreb index M ₁(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M ₂(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper we obtain an upper bound on the first Zagreb index M ₁(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ₁), second maximum vertex degree (Δ₂) and minimum vertex degree (δ). Using this result we find an upper bound on M ₂(G). Moreover, we present upper bounds on and in terms of n,  m, Δ₁, Δ₂, δ, where denotes the complement of G.     

Topological structures of tetrahedral (Td and T) fullerenes with hexagonal and triangular faces and their vibration and NMR spectra

On analyzing the topological structures of the three types of tetrahedral fullerenes (which consist only of triangles and hexagons), (1) C _n (T _d ,n=12h ²; h=1,2,…), (2) C _n (T _d ,n=4h ²;h=1,2,…), and (3) C _n (T,n=4(h ²+hk+k ²);h>k,h,k=1,2,…), we have obtained theoretically the Infrared and Raman active modes by means of the derived formulas for the decomposition of their nuclear motions into irreducible representations, and the ¹³C NMR spectra with natural abundance for ¹³C by using the distribution functions for all of the tetrahedral (T _d and T) fullerenes, respectively. Received: 25 May 1998 / Accepted: 30 July 1998 / Published online: 23 November 1998     

Synthesis and Spectroscopic Characterization of Two Different Types of Heterobimetallic Glycolate Complexes of Niobium(V)

An entirely new class of heterobimetallic homoleptic glycolate complexes of the type Nb(OGO)₃{Ta(OGO)₂} [where G=CMe₂CH₂CH₂CMe₂ (G¹) (3); CMe₂CH₂ CHMe(G²) (4); CHMeCHMe (G³) (5); CH₂CMe₂CH₂ (G⁴) (6); CMe₂CMe₂(G⁵) (7); CH₂CHMeCH₂ (G⁶) (8); CH₂CEt₂CH₂ (G⁷) (9); CH₂CMe(Prⁿ)CH₂ (G⁸) (10)] have been prepared by the reactions of Nb(OGO)₂(OGOH) [G=G¹ (1a); G² (1b); G³ (1c); G⁴ (1d); G⁵ (1e); G⁶ (1f); G⁷ (1g); G⁸ (1h)] with Ta(OGO)₂ (OPrⁱ) (G=G¹ (2a); G² (2b); G³ (2c); G⁴ (2d); G⁵ (2e) G⁶ (2f); G⁷ (2g); G⁸ (2h). In addition to the novel derivatives (2)–(10), our earlier investigations on heterobimetallic glycolate-alkoxide derivatives have been extended to derivatives of the type Nb(OGO) [where M=A1 n=3, G=G³ (11);G⁴ (12); G⁶ (13) G⁷ (14); G^s (15); G⁹=CH₂CH₂CH₂ (16) and M=Ti (n=4, G=G⁴) (17), Zr(n=4,G=G⁴) (18)], which are conveniently prepared by the reactions of metalloligands Nb(OGO)₂(OGOH) [G=G³ (1c); G⁴ (1d); G⁶ (1f); G⁷ (1g); G⁸ (1h); G⁹ (1i)] with different metal alkoxides. All of these new complexes have been characterized by elemental analyses, molecular weight determinations, and spectroscopic (I.r. and ¹H, ²⁷Al-n.m.r.) studies. Structural features of the new derivatives have been elucidated on the basis of molecular weight and spectroscopic data.     

Linking Tribridged Dicopper Complexes with Neutral Dipyridyl Compounds to Coordination Oligomers and Polymers

The stable tribridged dicopper(I) carboxylate complexes [Cu₂(μ‐dppm)₂(μ‐O₂CR)]BF₄ (RCO₂ ⁻ = formate (OFc⁻), m1 ; acetate (OAc⁻), m2 ; benzoate (OBAc⁻), m3 ; o‐toluate (O2TAc⁻), m4 ; p‐toluate (O4TAc⁻), m5 ; 4‐phenylbutyrate (O4PBAc⁻), m6 ; 2‐nitrobenzoate (O2NBAc⁻), m7 ), abbreviated as MM, and neutral dipyridyl compounds (NN; NN = 4,4′‐bipyridine (bpy), 1,2‐bis(4‐pyridyl)ethane (bpa), trans ‐1,2‐bis(4‐pyridyl)ethylene (bpe), 4,4′‐trimethylenedipyridine (tmp)) can form dynamic equilibria in CH₂Cl₂. From the equilibrium mixtures containing MM and NN with MM/NN = 1:1, nine 2:1 oligomers ([( m1 )₂(μ‐bpy)](BF₄)₂ ( o1a (BF₄)₂), [( m3 )₂(μ‐bpe)](BF₄)₂ ( o3c (BF₄)₂), [( m3 )₂(μ‐tmp)](BF₄)₂ ( o3d (BF₄)₂), [( m4 )₂(μ‐bpe)](BF₄)₂ ( o4c (BF₄)₂), [( m5 )₂(μ‐bpy)](BF₄)₂ ( o5a (BF₄)₂), [( m5 )₂(μ‐tmp)](BF₄)₂ ( o5d (BF₄)₂), [( m6 )₂(μ‐bpa)](BF₄)₂ ( o6b (BF₄)₂), [( m7 )₂(μ‐bpy)](BF₄)₂ ( o7a (BF₄)₂), [( m7 )₂(μ‐bpa)](BF₄)₂ ( o7b (BF₄)₂)), one 2:3 oligomer ([{( m2 )(bpy)}₂(μ‐bpy)](BF₄)₂ ( o2a (BF₄)₂)), and five 1:1 polymers ([( m2 )(μ‐bpe)] _n (BF₄ ) _n ( p2c (BF₄ ) _n ), [( m2 )(μ‐tmp)] _n (BF₄ ) _n ( p2d (BF₄ ) _n ), [( m3 )(μ‐bpy)] _n (BF₄ ) _n ( p3a (BF₄ ) _n ), [( m3 )(μ‐tmp)] _n (BF₄ ) _n ( p3d (BF₄ ) _n ), [( m7 )(μ‐tmp)] _n (BF₄ ) _n ( p7d (BF₄ ) _n )) were obtained as single crystals, and their structures were determined by X‐ray crystallography. Both experimental and theoretical results support the presence of two oligomeric species, [{Cu₂(μ‐dppm)₂(μ‐O₂CR)}₂(μ‐NN)]²⁺ and [{Cu₂(μ‐dppm)₂(μ‐O₂CR)(NN)}₂(μ‐NN)]²⁺), in dynamic equilibrium. The oligomers (such as o3d (BF₄)₂) can serve as seeds to induce the formation of soluble coordination polymers as crystals (such as p3d (BF₄)_n ).     

Sequence-adapted molecular tensors: Algebraic methods and application to crystal field theory

Tensorial sets adapted to sequences of finite subgroups are applied to the crystal field problem, and a general method for generating sequence-adapted molecular tensors using finite group algebra is formulated. All subgroup sequences of the abstract finite group G(24), isomorphic to the octahedral, O, tetrahedral, T_d, and symmetric, S(4), groups are tabulated with explicit isomorphisms provided. The sequences fall into eight equivalence classes. A catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences. With this data it is possible to systematically generate tensorial sets adapted to any sequence of a realization of G(24). Unitary transformations which adapt conventional forms of first- and second-rank irreducible tensorial sets of the rotation group to the eight sequences of the octahedral group are provided. Forms suitable for use with magnetic fields are included. The problem of a d¹ ion in a trigonal crystal field is treated with sequence-adapted molecular tensors, and the utility of different sequences for descent in symmetry is discussed.     

The forcing number of toroidal polyhexes

The forcing number, denoted by f(G), of a graph G with a perfect matching is the minimum number of independent edges that completely determine the perfect matching of G. In this paper, we consider the forcing number of a toroidal polyhex H(p,q,t) with a torsion t, a cubic graph embedded on torus with every face being a hexagon. We obtain that f(H(p,q,t)) ≥ min{p,q}, and equality holds for p ≤ q or p > q and t∈{ 0,p−q,p−q + 1,..., p−1}. In general, we show that f(H(p,q,t)) is equal to the side length of a maximum triangle on H(p,q,t). Based on this result, we design a linear algorithm to compute the forcing number of H(p,q,t).     

Subduction coefficients for 〈2N/2–S, 12S〉 ↓ 〈2, 1〉 ⊗ 〈2, 1〉 of U(n) ↓ U(n1) ⊗ U(n2)

The determination of the subduction coefficients for states of the unitary group U(n) under the restrictions U(n) ↓ U(n₁) ? U(n₂) have been considered for the spin free states of many electron systems. Using the transformation properties of the tensor basis spanning the irreducible representation 〈2^N/2–S, 1^2S〉 of U(n) under the permutations of electron coordinates, a simple programmable procedure has been developed for the determination of these coefficients. The procedure has been illustrated using a simple example.     

Kinetic analysis based on the excluded-volume effect for radical polymerization

When the potential of average force based on the excluded volume affects the relative motions of the polymer radicals, the specific rate for bimolecular reaction between them can be approximated as k_t = const. (ns)^?a, where a = 0.153(2b ? 1), b being a constant in the Mark-Houwink equation, and n and s being degrees of polymerization. Introduction of such a rate into kinetic equation yields a relative molecular weight distribution: G(n) = (n/m)^2–2a exp {ph(m^1–a ? n^1–a)}, where m = (2/ph)^1/(1–a) is a degree of polymerization for the maximum in G(n) and ph is a parameter denoting kinetic character. Further, the relationship between polymerization rate R_p, monomer concentration [M], and initiator concentration [ε] is found to be: where σ is a parameter denoting primary radical termination and η and η? are viscosities for an arbitrary solvent and ?-solvent, respectively. These relationships are sufficiently applicable to the data obtained in the polymerizations of styrene and methyl methacrylate.     

Reentrant behavior of poly(vinyl alcohol)–borax semidilute aqueous solutions

 The reentrant behavior of Poly(vinyl alcohol) (PVA)–borax aqueous semidilute solutions with a PVA concentration of 20 g/l and borax concentrations varies from 0.0 to 0.20 M was investigated using dynamic light scattering (DLS) and dynamic viscoelastic measurements. Two (fast and slow modes) and three (fast, middle, and slow) relaxation modes of PVA semidilute aqueous solutions without and with the presence of borax, respectively, were observed from DLS measurements. The fast and middle relaxation modes were q ²-dependent (q is the scattering vector) characteristic of diffusive behavior; however, the slow modes were q ³-dependent, characteristic of intraparticle dynamics. The experimental results showed that the slow relaxation mode dominates the DLS relaxation. The DLS slow mode relaxation time, τ_s, and the viscoelastic modulus G′(ω) and G′′(ω) data had a similar trend and demonstrated reentrant behavior as the borax concentration was increased from 0.0 to 0.20 M, i.e. τ_s, G′(ω), and G′′(ω) fluctuated with increasing borax concentration. The excluded-volume effect of polymers, charge repulsion among borate ions bound on PVA molecules, and intermolecular cross-linking didiol–borate complexation caused an expansion of the polymer chain; however, the screening effect of free Na⁺ ions on the negative charge of the borate ions bound on PVA and intramolecular cross-linking didiol–borate complexation led to a shrinkage of the polymer chain. The reentrant behavior was the consequence of the balance between expansion and shrinkage of the PVA–borate complex. Received: 26 March 1999/Accepted in revised form: 3 September 1999     

The group-theoretical structure of the atomic g shell: connection with the alternating group <Emphasis Type="Italic">A</Emphasis><Subscript>6</Subscript> as <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(9)

The special projective linear groups PSL(2ℓ + 1) or L ₂(2ℓ + 1) of order 2ℓ(2ℓ + 1)(ℓ + 1) can be used to study atomic shells of electrons with angular momentum quantum number ℓ corresponding to the atomic p, d, f, and g shells for ℓ = 1, 2, 3, 4, respectively. For the atomic g shell the group L ₂(9) is isomorphic with the alternating group A ₆ on six objects of order 360 or the symmetry group of the 5-dimensional simplex, a 5-dimensional analogue of the tetrahedron with 6 vertices and 15 edges. This leads to the subgroup chain SO(9) ⊃ SO(5) ⊃ L ₂(9) for the atomic g shell analogous to the subgroup chain SO(7) ⊃ G ₂ ⊃ L ₂(7) ≈⁷ O for the atomic f shell. In the L ₂(9) group only the representations of spherical harmonics or sums thereof, Γ(Y_ℓ), with dimensions dim Γ(Y_ℓ) or dim Γ(Y_ℓ) ± 1 divisible by 9 are found to be individually reducible to irreducible representations (irreps) or sums of irreps of L ₂(9). This leads to term groupings such as S, PD, G, PF, DH, L, PK, DI, FH, M, FI, PO, DN, HK, R, etc., of increasing total dimension for the irreps of SO(9) for various g ⁿ configurations in the atomic g shell.     

Vibration spectra and NMR spectra of octahedral (Oh and O) fullerenes

On analyzing the topological structures of the three big types of octahedral fullerenes: (1) C_n(O_h, n=24h²; h=1, 2,…); (2) C_n(O_h, n=8h²; h=1, 2,…), and (3) C_n(O, n=8(h²+hk+k²); h>k, h, k=1, 2,…), we have obtained theoretically the infrared and Raman active modes by means of the derived formulas for the decomposition of their nuclear motions into irreducible representations and the NMR spectra by using the distribution functions for all of the octahedral (O_h and O) fullerenes, respectively. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 199–205, 1999     

The aromatic diboracyclopropenyl (diboriranyl) anion; CB2H3 −: An ab initio study

The most stable structure of CB₂H₃ ^–, as established computationally, is the aromatic diboracyclopropenyl (diboriranyl) anion (5), while open-chainC _2v, isomer H₂BCBH (7) is only 3 kcal/mol higher in energy at the QCISD(T)/6-311 +G^**//MP2/6-31+G^*+ZPE (HF/6-31 +G^*). The 47-kcal/mol barrier between cyclic,5, and open-chain,7, structures suggests that both of them may be observed. The aromatic stabilization energy of the diboriranyl anion (18 kcal/mol) is half the value in the isoelectronic cyclopropenium ion, C₃H₃ ⁺. The computed, by IGLO method (5a), and experimental (6a) chemical shifts,(¹³C) and(¹¹B), agree within 4 ppm range. The theoretical vibrational frequencies of the most stable isomers,5 and7, are presented for experimental verification of these species.     

Photochemical reactions of chromium hexacarbonyl with tetraalkyldiphosphine disulfides

New complexes [Cr(CO)₄(R₂P(S)P(S)R₂)] and [Cr₂(CO)₁₀(-R₂P(S)P(S)R₂)] (R = Me, Et, Pr ⁿ , Bu ⁿ ), (1a)–(1d) and (2a)–(2d) [(1a), R = Me; (1b), R = Et; (1c), R = Pr ⁿ ; (1d), R = Bu ⁿ ; (2a), R = Me; (2b), R = Et; (2c), R = Pr ⁿ ; (2d), R = Bu ⁿ ] have been prepared by the photochemical reaction of Cr(CO)₆ with R₂P(S)P(S)R₂ (R = Me, Et, Pr ⁿ and Bu ⁿ ) and characterized by elemental analyses, FT-i.r., ³¹P-[¹H]-n.m.r. spectroscopy and FAB-mass spectrometry. The spectroscopic data suggest cis-chelate bidentate coordination of the ligand in [Cr(CO)₄(R₂P(S)P(S)R₂)] and cis-bridging bidentate coordination of the ligand between two metals in [Cr₂(CO)₁₀(-R₂P(S)P(S)R₂)] (R = Me, Et, Pr ⁿ and Bu ⁿ ).     

Ordering chemical unicyclic graphs by Wiener polarity index

Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d _G (u, v) = 3}, where d _G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W _p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polarity index is given.     

A conjecture for some partial differential operators on L 2( R n)

OnX =L ²(R ⁿ), letQ = (Q ₁,Q ₂,…,Q _n) andP = (P ₁,P ₂, …,P _n) be the operators given by (Q _jf) (x) =x _jf(x),P _j = - i∂/∂x _j. For anyC ^∞ functionh:R ⁿ →R putH ₀ =h(P) andH =H ₀ + (1 +Q ²)^-δ, where δ > 1/2. By the method of scattering theory we prove thatH _ac, the absolutely continuous part ofH is unitarily equivalent toH ₀ when (a)n = 1 and (b) forn ≥ 2, whenh is in a large class of polynomials. It is conjectured that the results are true for any polynomialh. We use the techniques of Enss’ method and the idea of bound states for momentum.     

On unicyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C _g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U ⁰(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C _g(G) and there are some edges of E(G)\ M(G) in G\ C _g(G) for any G∈U ⁰(k). In this paper, we discuss the graphs with minimal and second minimal energies in U ^*(k) = U(k)\ U ⁰(k), the graph with minimal energy in U ⁰(k), and propose a conjecture on the graph with minimal energy in U(k).