Average distance and maximum induced forest

With the help of the Graffiti system, Fajtlowicz conjectured around 1992 that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced bipartite subgraph of G, denoted α₂(G). We prove a strengthening of this conjecture by showing that the average distance between two vertices of a connected graph G is at most half the maximum order of an induced forest, denoted F(G). Moreover, we characterize the graphs maximizing the average distance among all graphs G having a fixed number of vertices and a fixed value of F(G) or α₂(G). Finally, we conjecture that the average distance between two vertices of a connected graph is at most half the maximum order of an induced linear forest (where a linear forest is a union of paths). © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 31–54, 2009     

Weakly connected domination stable trees

A dominating set D ⊆ V(G) is a weakly connected dominating set in G if the subgraph G[D] _w = (N _G [D], E _w ) weakly induced by D is connected, where E _w is the set of all edges having at least one vertex in D. Weakly connected domination number _γw (_G) of a graph G is the minimum cardinality among all weakly connected dominating sets in G. A graph G is said to be weakly connected domination stable or just _γw -stable if _γw (G) = _{γ w} (G + e) for every edge e belonging to the complement Ḡ of G. We provide a constructive characterization of weakly connected domination stable trees.      

Note on a theorem of Bangert

We generalize Bangert’s non-hyperbolicity result for uniformly tamed almost complex structures on standard symplectic ℝ²ⁿ to asymptotically standard symplectic manifolds.     

Upper tails for triangles

With ξ the number of triangles in the usual (Erd?s‐Rényi) random graph G(m,p), p > 1/m and η > 0, we show (for some C_η > 0) This is tight up to the value of C_η. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 452–459, 2012     

Maximum common subgraph isomorphism algorithms for the matching of chemical structures

The maximum common subgraph (MCS) problem has become increasingly important in those aspects of chemoinformatics that involve the matching of 2D or 3D chemical structures. This paper provides a classification and a review of the many MCS algorithms, both exact and approximate, that have been described in the literature, and makes recommendations regarding their applicability to typical chemoinformatics tasks.     

一种新的共轭桥联双二茂铁衍生物的单层LB膜及其电催化效应

合成并表征了一种新的非两亲性共轭桥联双二茂铁衍生物，该化合物在空气／水界面上易形成较稳定的单分子膜，与花生酸形成的混合膜易组装成ＬＢ膜，紫外－可见吸收光谱表明：ＬＢ上膜分子中发生了聚集，扫描电镜观察了混合单层ＬＢ膜的表面形貌及膜的相分离。循环伏安研究表明：混合单层ＬＢ膜修饰电极对Ｆｅ（ＣＮ）／Ｆｅ（ＣＮ）离子对的电化学反应具有良好的电催化效应，对抗坏血酸电化学氧化具有明显的促进作用。     

A 16‐connected three‐dimensional hydrogen‐bonding network in tetrakis(4‐aminopyridinium) perylene‐3,4,9,10‐tetracarboxylate octahydrate

The novel title organic salt, 4C₅H₇N₂⁺·C₂₄H₈O₈⁴⁻·8H₂O, was obtained from the reaction of perylene‐3,4,9,10‐tetracarboxylic acid (H₄ptca) with 4‐aminopyridine (4‐ap). The asymmetric unit contains half a perylene‐3,4,9,10‐tetracarboxylate (ptca⁴⁻) anion with twofold symmetry, two 4‐aminopyridinium (4‐Hap⁺) cations and four water molecules. Strong N—H...O hydrogen bonds connect each ptca⁴⁻ anion with four 4‐Hap⁺ cations to form a one‐dimensional linear chain along the [010] direction, decorated by additional 4‐Hap⁺ cations attached by weak N—H...O hydrogen bonds to the ptca⁴⁻ anions. Intermolecular O—H...O interactions of water molecules with ptca⁴⁻ and 4‐Hap⁺ ions complete the three‐dimensional hydrogen‐bonding network. From the viewpoint of topology, each ptca⁴⁻ anion acts as a 16‐connected node by hydrogen bonding to six 4‐Hap⁺ cations and ten water molecules to yield a highly connected hydrogen‐bonding framework. π–π interactions between 4‐Hap⁺ cations, and between 4‐Hap⁺ cations and ptca⁴⁻ anions, further stabilize the three‐dimensional hydrogen‐bonding network.     

Linear Connectivity,Schwarz–Pick Lemma and Univalency Criteria for Planar Harmonic Mapping

In this paper, we first establish a Schwarz–Pick lemma for higher-order derivatives of planar harmonic mappings, and apply it to obtain univalency criteria. Then we discuss distortion theorems,Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.     

On the hamiltonicity of a planar graph and its vertex-deleted subgraphs

Tutte proved that every planar 4-connected graph is hamiltonian. Thomassen showed that the same conclusion holds for the superclass of planar graphs with minimum degree at least 4 in which all vertex-deleted subgraphs are hamiltonian. We here prove that if in a planar $n$-vertex graph with minimum degree at least 4 at least $n-5$ vertex-deleted subgraphs are hamiltonian, then the graph contains two hamiltonian cycles, but that for every there exists a nonhamiltonian polyhedral $n$-vertex graph with minimum degree at least 4 containing $cn$ hamiltonian vertex-deleted subgraphs. Furthermore, we study the hamiltonicity of planar triangulations and their vertex-deleted subgraphs as well as Bondy's meta-conjecture, and prove that a polyhedral graph with minimum degree at least 4 in which all vertex-deleted subgraphs are traceable, must itself be traceable.