Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

First-principles study on electronic structures of BaWO4 crystals containing F-type color centers

The electronic structures of BaWO₄ crystals containing F-type color centers are studied within the framework of the fully relativistic self-consistent Dirac-Slater theory, using a numerically discrete variational (DV-Xα) method. It is concluded that F and F⁺ color centers have donor energy level in the forbidden band. The optical transition energies are 2.449 and 3.101 eV, which correspond to the 507 and 400 nm absorption bands, respectively. It is predicted that 400-550 nm absorption bands originate from the F and F⁺ color centers in BaWO₄ crystals.     

Generalized bond orders

We introduce generalized bond orders defined in terms of weighted Kekule valence structures. The weights were determined by the contributions of linearly independent and minimal conjugated circuits in individual Kekule valence structure. When special values for the contributions of conjugated circuits of different size are assumed, one obtains quantities that show considerable similarity to the Pauling and the Clar's bond orders. Pauling bond orders are obtained when one assumes that all conjugated circuits make equal contribution to bond orders. © 1994 John Wiley & Sons, Inc.     

Collapsible Graphs and Hamiltonicity of Line Graphs

Thomassen conjectured that every 4-connected line graph is Hamiltonian. Chen and Lai (Combinatorics and Graph Theory, vol 95, World Scientific, Singapore, pp 53–69; Conjecture 8.6 of 1995) conjectured that every 3-edge connected and essentially 6-edge connected graph is collapsible. Denote D ₃(G) the set of vertices of degree 3 of graph G. For ${e = uv \in E(G)}$ , define d(e) = d(u) + d(v) ? 2 the edge degree of e, and ${\xi(G) = \min\{d(e) : e \in E(G)\}}$ . Denote by λ ^m (G) the m-restricted edge-connectivity of G. In this paper, we prove that a 3-edge-connected graph with ${\xi(G)\geq7}$ , and ${\lambda^3(G)\geq7}$ is collapsible; a 3-edge-connected simple graph with ${\xi(G)\geq7}$ , and ${\lambda^3(G)\geq6}$ is collapsible; a 3-edge-connected graph with ${\xi(G)\geq6}$ , ${\lambda^2(G)\geq4}$ , and ${\lambda^3(G)\geq6}$ with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected simple graph with ${\xi(G)\geq6}$ , and ${\lambda^3(G)\geq5}$ with at most 24 vertices of degree 3 is collapsible; a 3-edge-connected graph with ${\xi(G)\geq5}$ , and ${\lambda^2(G)\geq4}$ with at most 9 vertices of degree 3 is collapsible. As a corollary, we show that a 4-connected line graph L(G) with minimum degree at least 5 and ${|D_3(G)|\leq9}$ is Hamiltonian.     

On the edge-connectivity of graphs with two orbits of the same size

It is well known that the edge-connectivity of a simple, connected, vertex-transitive graph attains its regular degree. It is then natural to consider the relationship between the graph’s edge-connectivity and the number of orbits of its automorphism group. In this paper, we discuss the edge connectedness of graphs with two orbits of the same size, and characterize when these double-orbit graphs are maximally edge connected and super-edge-connected. We also obtain a sufficient condition for some double-orbit graphs to be λ^′-optimal. Furthermore, by applying our results we obtain some results on vertex/edge-transitive bipartite graphs, mixed Cayley graphs and half vertex-transitive graphs.     

A lower bound on the number of removable ears of 1-extendable graphs

Let G be a 1-extendable graph distinct from K₂ and C_2n. A classical result of Lovász and Plummer (1986) [5, Theorem 5.4.6] states that G has a removable ear. Carvalho et al. (1999) [3] proved that G has at least Δ(G) edge-disjoint removable ears, where Δ(G) denotes the maximum degree of G. In this paper, the authors improve the lower bound and prove that G has at least m(G) edge-disjoint removable ears, where m(G) denotes the minimum number of perfect matchings needed to cover all edges of G.     

The connectivity ofZ-transformation graphs of perfect matchings of hexagonal systems

LetH be a hexagonal system. TheZ-transformation graphZ(H) is a graph where the vertices are perfect matchings ofH and where two perfect matchings are joined by an edge provided their synimetric difference consists of six edges of a hexagon ofH. We prove that the connectivity ofZ(H) is equal to the minimum degree of vertices ofZ(H).Project supported by the National Natural Science Foundation of China.     

Connectivity of the Mycielskian of a digraph

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), which is called the Mycielskian of G. This work investigates the vertex connectivity and arc connectivity of the Mycielskian of a digraph D. This generalizes the recent results due to Balakrishnan and Raj [R. Balakrishnan, S.F. Raj, Connectivity of the Mycielskian of a graph, Discrete Math, 308 (2008), 2607–2610].     

Simultaneous Determination of Tramadol and Acetaminophen in Human Plasma by LC–ESI–MS

This study presents a high-performance liquid chromatography–electrospray ionization–mass spectrometric (LC–ESI–MS) method for the simultaneous determination of tramadol and acetaminophen in human plasma using phenacetinum as the internal standard. After alkalization with saturated sodium bicarbonate, both compounds were extracted from human plasma with ethyl acetate and were separated by HPLC on a Hanbon LiChrospher CN column with a mobile phase of 10 mM ammonium acetate buffer containing 0.5% formic acid–methanol (40:60, v/v) at a flow rate of 1 mL min⁻¹. Analytes were determined using electrospray ionization in a single quadrupole mass spectrometer. LC–ESI–MS was performed in the positive selected-ion monitoring (SIM) mode using target ions at [M+H]⁺ m/z 264.3 for tramadol, [M+H]⁺ m/z 152.2 for acetaminophen and [M+H]⁺ m/z 180.2 for phenacetinum. Calibration curves were linear over the range of 5–600 ng mL⁻¹ for tramadol and 0.03–16 μg mL⁻¹ for acetaminophen. The inter-run relative standard deviations were less than 14.4% for tramadol and 12.3% for acetaminophen. The intra-run relative standard deviations were less than 9.3% for tramadol and 7.9% for acetaminophen. The mean plasma extraction recovery for tramadol and acetaminophen were in the ranges of 82.7–85.9 and 83.6–85.3%. The method was applied to study the pharmacokinetics of a new formulation of tramadol/acetaminophen tablet in healthy Chinese volunteers.