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Abstract: Let be a nontrivial connected graph of order and let be an integer with . For a set of vertices of , let denote the maximum number of edge-disjoint trees in such that for every pair of distinct integers with . A collection of trees in with this property is called an internally disjoint set of trees connecting . Chartrand et al. generalized the concept of connectivity as follows: The -connectivity, denoted by , of is defined by , where the minimum is taken over all -subsets of . Thus , where is the connectivity of . For general , the investigation of is very difficult. We therefore focus on the investigation on in this paper. We study the relation between the connectivity and the -connectivity of a graph. First we give sharp upper and lower bounds of for general graphs , and construct two kinds of graphs which attain the upper and lower bound, respectively. We then show that if is a connected planar graph, then , and give some classes of graphs which attain the bounds. In the end we give an algorithm to determine for general graphs . This algorithm runs in a polynomial time for graphs with a fixed value of connectivity, which implies that the problem of determining for graphs with a small minimum degree or connectivity can be solved in polynomial time, in particular, the problem whether for a planar graph can be solved in polynomial time. [Copyright &y& Elsevier]

Abstract: Gutman et al. introduced the concepts of energy and Laplacian energy for a simple graph G, and furthermore, they proposed a conjecture that for every graph G, is not more than . Unfortunately, the conjecture turns out to be incorrect since Liu et al. and Stevanović et al. constructed counterexamples. However, So et al. verified the conjecture for bipartite graphs. In the present paper, we obtain, for a random graph, the lower and upper bounds of the Laplacian energy, and show that the conjecture is true for almost all graphs. [Copyright &y& Elsevier]

Abstract: The activated carbon with honeycomb-like morphology for electrochemical double layer capacitors (EDLCs) was prepared from wheat straw by carbonization and subsequent activation with KOH. The product presented micro- and mesoporous structure with the peak pore size of 2.1nm and high specific surface area of 2316m2/g. The electrochemical capacitance performances of the activated carbon as EDLC electrodes were characterized by using cyclic voltammetry (CV), alternating current (AC) impedance. The specific capacitance of the activated carbon reached 251.1F/g at 2mV/s scan rate in the electrolyte of MeEt3NBF4/AN. The thermal gravimetric analysis of AN impregnated carbon was explored as a new approach to investigate the effectivity of pore for the electrolytes. The relationships between specific capacitance and scan rate, surface area and pore diameter were discussed. [Copyright &y& Elsevier]

Abstract: It is experimentally shown that the stability of non-overflowing pneumatic foams, such as those used in the Bikerman foam stability test, is strongly dependent upon the gradient of humidity within the freeboard of the column. Growing foams can be prevented from exhibiting collapse, even at surfactant concentrations significantly lower than the critical micelle concentration, by saturating the air above the foam surface thereby preventing liquid evaporation. These observations have important implications for the performance of flotation devices, as well as instruments designed to measure the stability of aqueous foams. Meteorological data for two mining sites in Australia throughout a single year shows the variability of humidity, and therefore stability, of flotation froths. [Copyright &y& Elsevier]

Abstract: A simple numerical scheme for solving the equation of Ward and Tordai (1946) for the diffusion-controlled adsorption of non-ionic surfactants to interfaces is proposed and pseudo-code, as well as C++ source code, is provided. The scheme utilises the trapezium rule of numerical integration and the accuracy and robustness of the method is enhanced by the bisection method of root-finding. The scheme is efficient and flexible in that it can be used with any adsorption isotherm and is readily modified for solving the problem of adsorption onto a convex interface. This scheme is not suggested for the adsorption onto a concave interface and the confusions that have previously arisen in relation to this problem are discussed. [Copyright &y& Elsevier]

Abstract: The energy of a graph is defined as the sum of the absolute values of its eigenvalues. A connected graph of order is said to be hypoenergetic if . All connected hypoenergetic graphs with maximum degree have been characterized. In addition to the four (earlier known) hypoenergetic trees, we now show that complete bipartite graph is the only hypoenergetic cycle-containing hypoenergetic graph. By this, the validity of a conjecture by Majstorović et al. has been confirmed. [Copyright &y& Elsevier]

Abstract: Given a graph and a subgraph of , let be the minimum number for which any edge-coloring of with colors has a rainbow subgraph . The number is called the rainbow number of with respect to . Denote as a matching of size and as the set of all the -regular bipartite graphs with bipartition such that and . Let be given positive integers, where , and . We show that for every , . We also determine the rainbow numbers of matchings in paths and cycles. [Copyright &y& Elsevier]

Abstract: The Randić index of a graph is defined by , where is the degree of a vertex in and the summation extends over all edges of . Aouchiche, Hansen and Zheng proposed the following conjecture: For any connected graph on vertices with Randić index and girth , with equalities if and only if . This paper is devoted to giving a confirmative proof to this conjecture. [Copyright &y& Elsevier]

Abstract: LiMnO2 and 0.23Li2MnO3·0.77LiMnO2 were prepared by a convenient one-step solid-state reaction from MnO2 using glucose as organic carbon resource. The crystal structure and morphology of the as-prepared materials was examined by X-ray powder diffraction and field emission scanning electron microscopy, respectively. The ration of Li to Mn was determined by means of atomic absorption spectrometry and the Li/Mn molar ratio in the products was 1.23. The electrochemical properties were investigated by charge–discharge test and electrochemical impedance measurements. The prepared composite material presented an initial discharge capacity of 45mAhg–1 and a good cycling performance with reversible capacity of 218mAhg–1 after 30 cycles. On the basis of the experimental results, the discharge efficiency of this composite material more than 100% was also discussed. [Copyright &y& Elsevier]

Abstract: For a given graph and a positive , the rainbow number of , denoted by , is the minimum integer so that in any edge-coloring of with colors there is a copy of whose edges have distinct colors. In 2004, Schiermeyer determined for all . The case for smaller values of (namely, ) remained generally open. In this paper we extend Schiermeyer’s result to all plausible and hence determine the rainbow number of matchings. [Copyright &y& Elsevier]

Abstract: Given two graphs and , let denote the maximum number for which there is a way to color the edges of with colors such that every subgraph of has at least two edges of the same color. Equivalently, any edge-coloring of with at least colors contains a rainbow copy of , where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number is called the rainbow number of with respect to , and simply called the bipartite rainbow number of if is the complete bipartite graph . Erdős, Simonovits and Sós showed that . In 2004, Schiermeyer determined the rainbow numbers for all , and the rainbow numbers for all and . In this paper we will determine the rainbow numbers for all . [Copyright &y& Elsevier]

Abstract: For positive integers and , a conditional -coloring of a graph is a proper -coloring of the vertices of such that every vertex of degree in is adjacent to vertices with at least different colors. The smallest integer for which a graph has a conditional -coloring is called the th-order conditional chromatic number, and is denoted by . It is easy to see that conditional coloring is a generalization of traditional vertex coloring (the case ). In this work, we consider the complexity of the conditional colorability of graphs. Our main result is that conditional -colorability remains NP-complete when restricted to planar bipartite graphs with maximum degree at most 3 and arbitrarily high girth. This differs considerably from the well-known result that traditional 3-colorability is polynomially solvable for graphs with maximum degree at most 3. On the other hand we show that -colorability is polynomially solvable for graphs with bounded tree-width. We also prove that some other well-known complexity results for traditional coloring still hold for conditional coloring. [Copyright &y& Elsevier]

Abstract: The energy of a graph is the sum of the absolute values of the eigenvalues of the graph. In a paper [G. Caporossi, D. Cvetkovi, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984–996] Caporossi et al. conjectured that among all connected graphs with vertices and edges, the graphs with minimum energy are the star with additional edges all connected to the same vertices for , and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. The conjecture is proved to be true for in the same paper by Caporossi et al. themselves, and for by Hou in [Y. Hou, Unicyclic graphs with minimal energy, J. Math. Chem. 29 (2001) 163–168]. In this paper, we give a complete solution for the second part of the conjecture on bipartite graphs. Moreover, we determine the graph with the second-minimal energy in all connected bipartite graphs with vertices and edges. [Copyright &y& Elsevier]

Abstract: Let be a simple undirected graph. Denote by (respectively, ) the number of maximal (respectively, maximum) independent sets in . Erdős and Moser raised the problem of determining the maximum value of among all graphs of order and the extremal graphs achieving this maximum value. This problem was solved by Moon and Moser. Then it was studied for many special classes of graphs, including trees, forests, bipartite graphs, connected graphs, (connected) triangle-free graphs, (connected) graphs with at most one cycle, and recently, (connected) graphs with at most cycles. In this paper we determine the second largest value of and among all graphs of order . Moreover, the extremal graphs achieving these values are also determined. [Copyright &y& Elsevier]

Abstract: Uniform nitrogen-enriched carbon-encapsulated nickel nanospheres were obtained by novel solvothermal method. The phase and morphology of as-prepared sample were characterized by X-ray diffraction (XRD), field-emission scanning electron microcopy (FE-SEM), transmission electron microscopy (TEM) and high-resolution transmission electron microscopy (HRTEM). It was found that the uniform spheres were made of core–shell structure with nickel core surrounded by 10nm thickness nitrogen-enriched carbon shell. Elemental analysis (EA) revealed the content of nitrogen in the sample reaching 15wt% and magnetic measurement showed a ferromagnetic behavior of the sample. On the basis of the experimental results, the growth mechanism of core–shell structural nanospheres was discussed. This solvothermal approach provided facile routine to form nitrogen-enriched nanospheres. [Copyright &y& Elsevier]

Abstract: The general all-ones problem is defined as follows: Given a graph , where V and E denote the vertex-set and the edge-set of G, respectively. Denote C as an initial subset of V. The problem is to find a subset X of V such that for every vertex of the number of vertices in X adjacent to is odd while for every vertex of C the number is even. X is called a solution to the problem. When , this problem is the so-called all-ones problem. If a vertex is viewed to be adjacent to itself, the all-ones problem is addressed as the all-ones problem. The all-ones problem has been studied extensively. However, the all-ones problem has received much less attention. Unlike the all-ones problem, which has solutions for any graphs, the all-ones problem may not have solutions for many graphs, even for some very simple graphs like and . So, it becomes an interesting question to find polynomial time algorithms to determine if for a given tree the problem has solutions. And if it does, to find a solution to the minimum all-ones problem. In this paper we present two algorithms of linear time to solve the general all-ones problem for trees. The first one is good for counting the number of solutions if solutions do exist, and the second one is good for solving the minimum all-ones problem. Furthermore, we can modify the algorithm slightly to solve the general minimum all-ones problem. [Copyright &y& Elsevier]

Abstract: A 2nd-order conditional -coloring of a graph is a proper -coloring of the vertices of such that every vertex of degree at least 2 in will be adjacent to vertices with at least 2 different colors. The smallest number for which a graph can have a 2nd-order conditional -coloring is the 2nd-order conditional chromatic number, denoted by . In this paper, we investigate the 2nd-order conditional 3-colorings of claw-free graphs. First, we prove that it is -complete to determine if a claw-free graph with maximum degree 3 is 2nd-order conditionally 3-colorable. Second, by forbidding a kind of subgraphs, we find a reasonable subclass of claw-free graphs with maximum degree 3, for which the 2nd-order conditionally 3-colorable problem can be solved in linear time. Third, we give a linear time algorithm to recognize this subclass of graphs, and a linear time algorithm to determine whether it is 2nd-order conditionally 3-colorable. We also give a linear time algorithm to color the graphs in the subclass by 3 colors. [Copyright &y& Elsevier]

Abstract: In this paper, we construct a large number of good quantum codes of minimum distances five and six by Steane''s Construction. Our methods involve the study of the check matrices of binary extended BCH-codes, together with puncturing and combining such matrices. [Copyright &y& Elsevier]

Abstract: Multiple equilibrium stages can be engendered in foam fractionation, a process used for the enrichment of streams of proteins, by returning some of the foamate stream to the top of the column as external reflux liquor. However, it was recognised, 40 years ago that reflux could be autogenously created through the coalescence of bubbles in fractionation columns. By invoking the hydrodynamic theory of rising foam, we suggest a mechanism for the creation of internal reflux in foam fractionation. This method can give internal reflux rate as a function of bubble size. However, since the bubble size profile in a rising foam cannot be estimated, we cannot yet estimate how internal reflux varies with position in the column. [Copyright &y& Elsevier]

Abstract: The incidence coloring conjecture, proposed by Brualdi and Massey in 1993, states that the incidence coloring number of every graph is at most , where is the maximum degree of a graph. The conjecture was shown to be false in general by Guiduli in 1997, following the work of Algor and Alon. However, in 2005 Maydanskiy proved that the conjecture holds for any graph with . It is easily deduced that the incidence coloring number of a semi-cubic graph is 4 or 5. In this paper, we show that it is already NP-complete to determine if a semi-cubic graph is 4-incidence colorable, and therefore it is NP-complete to determine if a general graph is k-incidence colorable. [Copyright &y& Elsevier]

Abstract: Let be the eigenvalues of a graph of order . The energy of is defined as . Let be the graph obtained from two copies of joined by a path , be the class of all bipartite bicyclic graphs that are not the graph obtained from two cycles and ( and (mod 4)) joined by an edge. In this paper, we show that is the graph with maximal energy in , which gives a partial solution to Gutman’s conjecture in Gutman and Vidović (2001) [I. Gutman, D. Vidović, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41 (2001) 1002–1005]. [Copyright &y& Elsevier]

Abstract: Novel ordered nickel chains with diameter 250–500nm and length more than 5μm were synthesized under weak magnetic field by hydrazine reducing in ethylene glycol. The phase, morphology and magnetic properties were characterized by X-ray diffraction (XRD), scanning electron microanalyzer (SEM) and vibrating sample magnetometer (VSM), respectively. The results reveal that weak magnetic field leads to the fabrication of nickel chains paralleling each other with hierarchical structures. The growth mechanism and fabrication process of nickel magnetic nanocrystallites were discussed. [Copyright &y& Elsevier]

Abstract: Highly ordered nanorods array of B phase vanadium dioxide was firstly synthesized with n-butanol as the reducing agent via a simple hydrothermal method without using template. The samples have been characterized by X-ray diffraction (XRD), transmission electron microscopy (TEM), and field emission scanning electron microscopy (FE-SEM). The size of VO2 (B) nanorods has the dimension of 50–100nm in diameter and about 1–5μm in length. The samples were measured as electrode materials by charge–discharge technique and the VO2 (B) nanorods array demonstrated a high specific capacity of 520mAh/g at 0.2C. The influence of reaction temperature on fabricating nanorods array has been studied. The possible growth mechanisms of formation of nanorods and assembly of array were discussed. [Copyright &y& Elsevier]

Abstract: The minimum all-ones problem and the connected odd dominating set problem were shown to be NP-complete in different papers for general graphs, while they are solvable in linear time (or trivial) for trees, unicyclic graphs, and series-parallel graphs. The complexity of both problems when restricted to bipartite graphs was raised as an open question. Here we solve both problems. For this purpose, we introduce the related decision problem of the existence of an odd dominating set without isolated vertices, and study its complexity. Our main result shows that this new problem is NP-complete, even when restricted to bipartite graphs. We use this result to deduce that the minimum all-ones problem and the connected odd dominating set problem are also NP-complete for bipartite graphs. We show that all three problems are solvable in linear time for graphs with bounded treewidth. We also show that the new problem remains NP-complete when restricted to other graph classes, e.g., planar graphs, graphs with girth at least five, and graphs with a small maximum degree, in particular 3-regular graphs. [Copyright &y& Elsevier]

Abstract: Let be an edge-colored graph. A subgraph is said to be monochromatic if all the edges of have the same color, and multicolored if no two edges of have the same color. We investigate the complexity of the problems for finding the minimum number of monochromatic or multicolored subgraphs, such as cliques, cycles, trees and paths, partitioning , depending on the number of colors used and the maximal number of times a color appears in a coloring. We also present a greedy scheme that yields a -approximation for the problem of finding the minimum number of monochromatic cliques partitioning for a -free graph , where is the size of the largest monochromatic clique in . By a slightly modification of the approximation algorithm, it can be used for the multicolored case. We show that unless , for any there is no approximation algorithm for finding the minimum number of multicolored trees partitioning with performance . [Copyright &y& Elsevier]

Abstract: The Randić index of a (chemical) graph G is also called connectivity index. Hansen and Mélot [Variable neighborhood search for extremal graphs 6: analyzing bounds for the connectivity index, J. Chem. Inf. Comput. Sci. 43 (2003) 1–14] in their paper, characterized the chemical trees of given order and number of pendent vertices which have the minimum and maximum Randić index, respectively. In this note, we point out the mistakes in the proofs of their results Theorems 8 and 10, while we still believe that the two theorems are true, and then we give their corrected proofs. [Copyright &y& Elsevier]

Abstract: Neighbour-scattering number is a useful measure for graph vulnerability. For some special kinds of graphs, explicit formulas are given for this number. However, for general graphs it is shown that to compute this number is NP-complete. In this paper, we prove that for interval graphs this number can be computed in polynomial time. [Copyright &y& Elsevier]

Abstract: The adsorption kinetics of Th(IV) on oxidized multi-wall carbon nanotubes (MWCNTs) was investigated at initial Th(IV) concentrations 32.32 and 64.64μmol/L. The kinetic process was described by a pseudo-second-order rate model very well. The adsorption thermodynamics of Th(IV) on oxidized MWCNTs was carried out at 293±2, 303±2, and 323±2K, respectively, and the thermodynamic parameters, such as equilibrium constant (K 0), standard free energy changes (ΔG°), standard enthalpy change (ΔH°) and standard entropy change (ΔS°), were obtained. The Langmuir model described the data better than the Freundlich isotherm model. Desorption studies indicated that Th(IV) adsorption was reversible and Th(IV) ions could be desorbed from the surface of carbon nanotubes by adjusting the pH values. Oxidized MWCNTs may be a promising candidate for the preconcentration and solidification of Th(IV), or its analogue actinides from large volumes. [Copyright &y& Elsevier]

Abstract: Spinel LiMn2O4 crystal with the grain sizes of about 15 nm is firstly synthesized by hydrothermal route at 180 °C using MnO2 as a precursor. The LiMn2O4 powders synthesized by hydrothermal technique and sol–gel reaction were investigated by X-ray diffraction (XRD) and Transmission electron microscopy (TEM). The LiMn2O4 samples were used as cathode materials for lithium-ion battery, whose electrochemical properties were investigated. The results show that the sample obtained by hydrothermal route has higher capacity than that prepared by sol–gel method. [Copyright &y& Elsevier]

Abstract: Let G be a graph and denote the degree of a vertex in G. Then the zeroth-order general Randić index of the graph G is defined as , where is a pertinently chosen real number. We characterize, for any , the connected -graphs with minimum and maximum . [Copyright &y& Elsevier]

Abstract: Flowerlike ZnO nanostructures were prepared by a PEG2000 and urea-assisted hydrothermal process at 140 °C. The samples were characterized by X-ray diffraction, transmission electron microscopy and photoluminescence spectrum. Results show that urea played a significant role in the formation of ZnO nanostructures. A preliminary explanation of the mechanism is proposed. [Copyright &y& Elsevier]

We give elementary recursive constructions of quaternary self-orthogonal codes with dual distance three for all n ≥ 5. Consequently, good linear quantum codes of minimum distance three for such length n are obtained. Almost all of these linear quantum codes are optimal or near optimal. [ABSTRACT FROM AUTHOR]

In this paper, we first reduce the problem of finding a minimum parity ( g, f)-factor of a graph G into the problem of finding a minimum perfect matching in a weighted simple graph G*. Using the structure of G*, a necessary and sufficient condition for the existence of an even factor is derived. [ABSTRACT FROM AUTHOR]

Abstract: Zinc-doped SnO2 nanocrystals were prepared by a simple hydrothermal process. X-ray diffraction (XRD) patterns show that as-prepared samples are of tetragonal rutile structure of SnO2. Transmission electron microscopy (TEM) and field emission scanning electron microscopy (FE-SEM) characterizations confirm that the morphologies of zinc-doped SnO2 nanocrystals are “flower”-constructed nanorods. Photoluminescence (PL) measurement of the zinc-doped SnO2 crystals reveals a purple emission band centered at ∼440nm. The influences of the hydrothermal reaction temperature, the molar ratio of Zn2+ and Sn4+ as well as the concentrations of NaOH have been studied. The possible growth mechanisms of flower-like nanocrystals were discussed. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the k-path cover problem for graphs, which is to find the minimum number of vertex disjoint k-paths that cover all the vertices of a graph. The k-path cover problem for general graphs is NP-complete. Though notable applications of this problem to database design, network, VLSI design, ring protocols, and code optimization, efficient algorithms are known for only few special classes of graphs. In order to solve this problem for cacti, i.e., graphs where no edge lies on more than one cycle, we introduce the so-called Steiner version of the k-path cover problem, and develop an efficient algorithm for the Steiner k-path cover problem for cacti, which finds an optimal k-path cover for a given cactus in polynomial time. [Copyright &y& Elsevier]

Abstract: Arrays of NiO nanoparticles have been synthesized via thermal decomposition of nickel dimethylglyoximate precursor. The structure, morphology and properties of the products were characterized by XRD, TEM, SAED and UV–vis spectrophotometry. TEM reveals that one-dimensional arrays consist of NiO nanoparticles. XRD and SAED indicate that these NiO nanoparticles crystallize with a polycrystalline structure. The optical absorption band gap of NiO nanoparticles is 3.51eV. [Copyright &y& Elsevier]

Abstract: In this paper, some new families of integral trees with diameters 5 and 6 are constructed. All these classes are infinite. These are different from those in the existing literature. We also prove that the problem of finding integral trees of diameters 5 and 6 is equivalent to the problem of solving some Diophantine equations. The discovery of these integral trees is a new contribution to the search for such trees. [Copyright &y& Elsevier]

Abstract: For a graph G, let denote its adjoint polynomial and denote the minimum real root of . Two graphs H and G are said to be adjointly equivalent if . Let and . In this paper, we give a necessary and sufficient condition for two graphs H and G in to be adjointly equivalent, where . We also solve some problems and conjectures proposed by Dong et al. (Discrete Math. 258 (2002) 303–321). [Copyright &y& Elsevier]

In this paper, we introduce the concept of sequential dynamical systems (SDS) on digraphs. We focus on the discussion of linear sequential dynamical systems (LSDS). Matrix method is given in their analysis. Two special LSDS, OR-SDS and PAR-SDS, are particularly analyzed. Some structural properties on the image spaces of [ORD,π] and [PARD,π] are obtained. The asymptotic behavior of [ORD,π] is described in terms of the properties of the digraph D with respect to the ordering π. Our results show that LSDS on digraphs have much more interesting properties than those on undirected graphs. [Copyright &y& Elsevier]

Abstract: For graphs G and with minimum degree at least k, when , we prove that any isomorphism from the -graph onto can be induced by a (vertex-)isomorphism of G onto . [Copyright &y& Elsevier]

Let G be an edge-coloured graph. We show in this paper that it is NP-hard to find the minimum number of vertex disjoint monochromatic trees which cover the vertices of the graph G . We also show that there is no constant factor approximation algorithm for the problem unless P = NP. The same results hold for the problem of finding the minimum number of vertex disjoint monochromatic cycles (paths, respectively) which cover the vertices of the graph. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting in the graph ; second, for all the vertices x of degree 3 in , delete x from and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by . If is still 4-connected, then e is called a removable edge of G. In this paper, we investigate the problem on how many removable edges there are in a cycle of a 4-connected graph, and give examples to show that our results are in some sense the best possible. [Copyright &y& Elsevier]

LetGbe a 4-connected graph. For an edgeeofG, we do the following operations onG: first, delete the edgeefromG, resulting the graphG-e; second, for all verticesxof degree 3 inG-e, deletexfromG-eand then completely connect the 3 neighbors ofxby a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted byG?e. IfG?eis 4-connected, theneis called a removable edge ofG. In this paper we obtain some results on removable edges in a longest cycle of a 4-connected graphG. We also show that for a 4-connected graphGof minimum degree at least 5 or girth at least 4, any edge ofGis removable or contractible. [ABSTRACT FROM AUTHOR]

Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G-e; second, for all the vertices x of degree 3 in G-e, delete x from G-e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G⊖e. If G⊖e is still 4-connected, then e is called a removable edge of G. In this paper we prove that every 4-connected graph of order at least six (excluding the 2-cyclic graph of order six) has at least (4|G|+16)/7 removable edges. We also give the structural characterization of 4-connected graphs for which the lower bound is sharp. [Copyright &y& Elsevier]

A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we give a useful sufficient and necessary condition for complete r-partite graphs to be integral, from which we can construct infinite many new classes of such integral graphs. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving some Diophantine equations. The discovery of these integral complete r-partite graphs is a new contribution to the search of such integral graphs. Finally, we propose several basic open problems for further study. [Copyright &y& Elsevier]

We give elementary recursive constructions of binary self- orthogonal codes with dual distance four for all even lengths n ≥ 12 and n = 8. Consequently, good quantum codes of minimum distance three and four for such length n are obtained via Steane's construction and the CSS construction. Previously, such quantum codes were explicitly constructed only for a sparse set of lengths. Almost all of our quantum codes of minimum distance three are optimal or near optimal, and some of our minimum-distance four quantum codes are better than or comparable with those known before. [ABSTRACT FROM AUTHOR]

Let β(G) denote the minimum real root of the σ-polynomial of the complement of a graph G and δ(G) the minimum degree of G. In this paper, we give a characterization of all connected graphs G with β(G)⩾−4. Using these results, we establish a sufficient and necessary condition for a graph G with p vertices and δ(G)⩾p−3, to be chromatically unique. Many previously known results are generalized. As a byproduct, a problem of Du (Discrete Math. 162 (1996) 109–125) and a conjecture of Liu (Discrete Math. 172 (1997) 85–92) are confirmed. [Copyright &y& Elsevier]

Some new families of integral trees with diameters 4, 6 and 8 are given. Most of these classes are infinite. [Copyright &y& Elsevier]

For a set C of cycles of a connected graph G we define T(G,C) as the graph with one vertex for each spanning tree of G, in which two trees R and S are adjacent if R∪S contains exactly one cycle and this cycle lies in C. We give necessary conditions and sufficient conditions for T(G,C) to be connected. [Copyright &y& Elsevier]

We consider the following special case of a conjecture due to Caccetta and Ha¨ggkvist: Let D be a digraph on n vertices that all have in-degree and out-degree at least n/3. Then, D contains a directed cycle of length 2 or 3. We discuss several necessary conditions for possible counterexamples to this conjecture, in terms of cycle structure, diameter, maximum degree, clique number, toughness, and local structure. These conditions have not enabled us to prove or refute the conjecture, but they lead to proofs of special instances of the conjecture. [Copyright &y& Elsevier]