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.     

Sharp upper bounds for total π-electron energy of alternant hydrocarbons

The energy E(G) of a graph G is defined as the sum of the absolute values of all the eigenvalues of the adjacency matrix of the graph G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. In this paper, we show that if G = (V ₁, V ₂; E) is a bipartite graph with edges and , then

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

Potential energy function based on the narcissus constant,its square and its cube

The narcissus constant, N = 2.3983843828..., is defined as a number that fulfills the narcissistic infinite nested radical equation

Zero point energy of the Pullen–Edmonds Hamiltonian

Here we wish to apply the newly developed Generalized Moments Expansion (GMX) to the well-known potential

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.     

Extremal Polyomino Chains on k-matchings and k-independent Sets

Denote by the set of polyomino chains with n squares. For any , let m _k (T _n ) and i _k (T _n ) be the number of k-matchings and k-independent sets of T _n , respectively. In this paper, we show that for any polyomino chain and any , and , with the left equalities holding for all k only if T _n =L _n , and the right equalities holding for all k only if T _n =Z _n , where L _n and Z _n are the linear chain and the zig-zag chain, respectively. This work is supported by NNSFC (10371102).     

Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let be the graph obtained by identifying the center of the star S _n-l+1 with any vertex of C _l . By we denote the graph obtained by identifying one pendent vertex of the path P _n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).      

Calorimetric search for reliable excess enthalpy data as a function of temperature

In the search of a useful method for determining excess enthalpies as a function of temperature Calvet calorimetry was employed. To this end, excess molar enthalpies

On zeroth-order general Randić index of conjugated unicyclic graphs

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as where α is an arbitrary real number. In this paper, we investigate the zeroth-order general Randić index of conjugated unicyclic graphs G (i.e., unicyclic graphs with a perfect matching) and sharp lower and upper bounds are obtained for depending on α in different intervals.     

Nickel(II) salophen-type complexes characterization and their thermodynamic studies with diorganotin(IV) dichlorides in chloroform

A u.v.–vis spectrophotometric study of the adduct formation of the nickel(II) Schiff base complexes,([NiL]) where L = [3-methoxysalophen, N,N′-bis(3-methoxysalicylidene)-1,2-phenylenediimine] (1), [4-methoxysalophen, N,N′-bis(4-methoxysalicylidene)-1,2-phenylenediimine] (2), [5-methoxysalophen, N,N′-bis(5-methoxysalicylidene)-1,2-phenylenediimine] (3) and [Salophen, [N,N′-bis(salicylaldehydo)-1,2-phenylenediimine] (4) as donors with R₂SnCl₂ (R = methyl, phenyl and n-butyl) as acceptors have been investigated in chloroform, as solvent. Adducts have been characterized by ¹H, ¹³C and ¹¹⁹Sn NMR, IR and electronic spectroscopy and CHN elemental microanalysis. The formation constants and the thermodynamic free energies were measured using u.v.–vis spectrophotometry titration for 1:1 adduct formation at various temperatures (T = 278 to 308 K). The trend of the adduct formation of the nickel Schiff base complexes with a given tin acceptor decreases as follows:

The System HCl + NdCl3 + H2O from 5 to 55 ∘C: A Study of Harned's Rule

The activity coefficients of HCl (γ_A) in aqueous mixtures of HCl and NdCl₃ were determined by the electromotive-force (emf) measurement of cells without liquid junctions of the type:

Unicycle graphs with extremal Merrifield–Simmons Index

The Merrifield–Simmons index f(G) of a (molecular) graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex sets of G. By we denote the set of unicycle graphs in which the length of its unique cycle is k. In this paper, we investigate the Merrifield–Simmons index f(G) for an unicycle graph G in . Unicycle graphs with the largest or smallest Merrifield–Simmons index are uniquely determined.     

Unicycle Graphs with Maximum Generalized Topological Indices

Let G be an unicycle graph and d _v the degree of the vertex v. In this paper, we investigate the following topological indices for an unicycle graph , , where m ≥ 2 is an integer. All unicycle graphs with the largest values of the three topological indices are characterized. This research is supported by the National Natural Science Foundation of China(10471037)and the Education Committee of Hunan Province(02C210)(04B047).     

Thermodynamic Study of the NaCl + Na2SO4 + H2O System by Emf Measurements at Four Temperatures

Activity coefficients for sodium chloride in the NaCl + Na₂SO₄ + H₂O ternary system were determined from emf measurements of the cell

Solubility of Oxygen in Some 1-1, 2-1, 1-2, and 2-2 Electrolytes as a Function of Concentration at 25°C

The solubility of oxygen has been measured in a number of electrolytes [(LiCl, KCl, RbCl, CsCl, NaF, NaBr, NaI, NaNO₃, KBr, KI, KNO₃, CaCl₂, SrCl₂, BaCl₂, Li₂SO₄, K₂SO₄, Mn(NO₃)₃)] as a function of concentration at 25°C. The solubilities, mol (kg-H₂O)^–1, have been fitted to a function of the molality m (standard deviation < 3mol-kg^–1)

On the path-Zagreb matrix

The definition of the path-Zagreb matrix for (chemical) trees PZ and its generalization to any (molecular) graph is presented. Additionally, the upper bound of , where G _n is a graph with n vertices is given.     

Studies on the Kinetics of the First Order Autocatalytic Decomposition Reaction of Highly Nitrated Nitrocellulose

The kinetics of the first order autocatalytic decomposition reaction of highly nitrated nitrocellulose (HNNC, 14.14%N) was studied by using thermogravimetry (TG). The results show that the TG curve for the initial 50% of mass-loss of HNNC can be described by the first order autocatalytic equation

The thermodynamics of aqueous borate solutions. II. Mixtures of boric acid with calcium or magnesium borate and chloride

Potentials for the cell without liquid juction

FT-IR Investigation of the State of Iron(III) Chloride Clusters Confined in AOT Reverse Micelles Dispersed in Carbon Tetrachloride

The state of the water-soluble salt iron(III) chloride in AOT reverse micelles dispersed in carbon tetrachloride has been investigated by FT-IR spectroscopy. Interestingly, while the entrapment of a lot of water-soluble inorganic salts in AOT reverse micelles requires preliminarily the presence of significant amounts of water within the micellar core, solubilization of FeCl₃ occurs without the need to add water in the micellar solution reaching the very high solubility value, expressed as the maximum salt-to-surfactant molar ratio, of 1.30. The analysis of the spectral features of the investigated samples leads to hypothesize that iron(III) chloride is confined within the reverse micellar core as small size melted clusters of ionic species arising from the reactions