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Let $\Phi=(G,U(\mathbb{Q}),\varphi)$ be a quaternion unit gain graph (or $U(\mathbb{Q})$-gain graph), where $G$ is the underlying graph of $\Phi$, $U(\mathbb{Q})=\{z\in \mathbb{Q}: |z|=1\}$ is the circle group, and $\varphi:\overrightarrow{E}\rightarrow U(\mathbb{Q})$ is the gain function such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$. Let $A(\Phi)$ be the adjacency matrix of $\Phi$ and $r(\Phi)$ be the row left rank of $\Phi$. In this paper, we prove that $-2c(G)\leq r(\Phi)-r(G)\leq 2c(G)$, where $r(G)$ and $c(G)$ are the rank and the dimension of cycle space of $G$, respectively. All corresponding extremal graphs are characterized. The results will generalize the corresponding results of signed graphs (Lu et al. [20] and Wang [33]), mixed graphs (Chen et al. [7]), and complex unit gain graphs (Lu et al. [21]).

MoS2 nanosheets are grown in situ onto hollow MnS microcubes to produce 3D hierarchical bimetallic sulfide architecture (MnS@MoS2) for supercapacitors. The asymmetric supercapacitor device assembled using MnS@MoS2 achieves energy density of 64.98 W h kg−1 at 1600 W kg−1.

Non-noble metal electrocatalysts have attracted great attention on account of low cost, good stability, and excellent hydrogen evolution performance. One of the most efficient approaches to increas...

Purpose Creativity is highly valued by modern society; however, in China, there is a major challenge in creating school-based curricula or courses that promote students’ creativity. This study introduced an interactive learning format called Collaborative Reasoning (CR). The goal was to investigate whether CR would improve a student's ability to generate creative ideas. Design/Approach/Methods The data comprised 474 seventh graders from eight classes in a rural middle school in northern China. The eight classes were randomly assigned to two conditions: Four classes participated in CR discussions, and the remaining four classes were taught via teacher-directed Chinese reading classes. After participating in the CR discussions, students completed a Realistic Presented Problem test that was designed to evaluate their creative performance. Findings The results indicated that compared to the control students, CR students generated more effective solutions (fluency) and proposed a greater variety of solutions (flexibility) to three realistic problems. The solutions proposed by the CR students were more unique (novelty) and more useful in real-life situations (usefulness) than those proposed by the control group. Originality/Value Providing abundant opportunities for the exchange of ideas in a social context can boost students’ performance in creative problem-solving.

In this paper, we present sufficient conditions on eccentric connectivity index, eccentric distance sum and connective eccentricity index for graphs to be k-hamiltonian, k-edge-hamiltonian or k-path-coverable.

In this paper, we present two new sufficient conditions on the spectral radius ρ ( G ) that guarantee the hamiltonicity and traceability of a k-connected graph G of sufficiently large order, respectively, unless G is a specified exceptional graph. In particular, if k ≥ 2 , n ≥ k 3 + k + 2 , and ρ ( G ) > n − k − 1 − 1 n , then G is hamiltonian, unless G is a specified exceptional graph. If k ≥ 1 , n ≥ k 3 + k 2 + k + 3 , and ρ ( G ) > n − k − 2 − 1 n , then G is traceable, unless G is a specified exceptional graph.

In this paper, we present some spectral sufficient conditions for a graph to be Hamilton-connected in terms of the spectral radius or signless Laplacian spectral radius of the graph. Our results improve some previous work.

In this paper, we present a spectral sufficient condition for a graph to be Hamilton-connected in terms of signless Laplacian spectral radius with large minimum degree.

The synthesis of highly active, stable and cost-effective electrocatalysts for overall water splitting is critical but challenging. Herein, we synthesized nickel-doped pyrrhotite iron sulfide (Fe0.95−xNixS1.05) nanosheets via a one-pot synthetic route. When tested for oxygen evolution reaction (OER) performance in alkaline solutions, the as-prepared Fe0.8Ni0.15S1.05 catalyst exhibited superior electrocatalytic performance to commercial IrO2 with a current density of 10 mA cm−2 at an overpotential of 228 mV, and a Tafel slope of 53 mV dec−1. The hydrogen evolution reaction (HER) activity was achieved on Fe0.8Ni0.15S1.05 with a Tafel slope of 103 mV per decade, and a current density of 10 mA cm−2 at an overpotential of 263 mV. With nickel doping, the as-prepared pyrrhotite iron sulfide nanosheets can expose more active sites, and hence act as a highly efficient catalyst for overall water splitting.

In this study, cellulose-based superabsorbent hydrogel was synthesized from sodium carboxymethyl cellulose (CMC-Na), acrylic acid (AA), and 2-acrylamido-2-methylpropanesulfonic acid (AMPS) to enhance its water absorbency and salt tolerance for soil-conditioning applications in areas suffering from drought and soil salinization. Superabsorbent hydrogels (SHs) were prepared by CMC-Na and AMPS successfully, using chemical graft technology. Structure, morphology, thermal stability, and water absorbency of SHs were deduced. The cellulose-based hydrogel showed a high salt tolerance that the maximum water absorbency reached 604 and 119% in distilled water and saline water, respectively. The swelling behavior in aqueous solvents indicated that the water absorption of hydrogels was improved with the increasing ratio of CMC-Na. All SHs exhibited adsorption of nitrogen with the maximum adsorption of ammonia nitrogen 30 mg·g

We present two new sufficient conditions in terms of the spectral radius $$\rho (G)$$ ρ ( G ) guaranteeing that a k-connected graph G is Hamilton-connected, unless G belongs to a collection of exceptional graphs. We use the Bondy–Chvátal closure to characterize these exceptional graphs.

Superwetting interfacial materials have been broadly developed for extensive applications. Reversible superwetting behavior has received a great deal of attention because of the potential applications. In this paper, a hierarchical CuO coating on a stainless steel substrate was fabricated simply by a solution-immersion process at 85 °C. The as-prepared CuO-coated stainless steel surface was superhydrophilic, and obtained superhydrophobicity by modification with tetradecanoic acid for only 15 s. The modified surface converted back into superhydrophilic after being annealed in muffle furnace at 300 °C for 5 min. Moreover, the annealed surface became superhydrophobic again by remodification. A rapid reversible superwetting transition cycle was finished by modification with tetradecanoic acid and annealing treatment for only 5.25 min. When the reversible superwetting transition was applied to the CuO-coated stainless steel mesh, the resulting mesh was switchable between “oil-removing” and “water-removing” modes. Thus, the mesh was suitable for oil/water mixtures whether heavy oil or light oil is involved. After 10 cycles of the superwetting transition or separation for 10 times, the separation efficiency of both meshes remained over 99.9%. The meshes showed the stability and recyclability.

The Wiener-type invariants of a simple connected graph G = (V(G), E(G)) can be expressed in terms of the quantities Wf = ? {u,v}?V(G)f(dG(u,v)) for various choices of the function f(x), where dG(u,v) is the distance between vertices u and v in G. In this paper, we give some sufficient conditions for a bipartite graph to be Hamiltonian or a connected general graph to be Hamilton-connected and traceable from every vertex in terms of the Wiener-type invariants of G or the complement of G.

Numerous superwetting materials have been designed to separate water-in-oil emulsions, however, it still remains challenge to realize emulsion separation and dye adsorption simultaneously. In this work, we adopt a simple strategy to remove dyes and tiny water droplets from the dyed surfactant-stabilized water-in-oil emulsion using layer-stacked waste honeycomb cinder (WHC). The layer-stacked WHC with under-oil superhydrophilicity is obtained to separate the dyed emulsions only driven by gravity. The volumetric water fraction of the emulsion is 1%. The WHC displays exceptional separation efficiency and high filtrate flux for the emulsion separation, and highly efficient adsorption for various kinds of dyes in water, oil, and emulsion. Interestingly, no tiny water droplets in the filtrate are detected by dynamic light scattering (DLS), even though the concentration of surfactant in the emulsion is up to 2.4 mg/mL. Furthermore, the WHC could remain considerable recyclability after 10 cycles of emulsion separation. The separation efficiency could be improved after the WHC is immersed in corrosive liquid, which shows the excellent chemical durability of the WHC. This work demonstrates that the WHC as green and sustainable material can be efficiently reused to separate emulsions and adsorb dyes.

In recent years, great attention has been devoted to transition metal-based dichalcogenides (TMDs), in particular to the 1 T-phase enriched TMDs, which possess much higher hydrogen evolution reaction (HER) activity than the 2H-phase. Currently, how to prepare the stable metallic phase TMDs nanomaterials to achieve an endurable HER is one of the main challenges for practical applications. In this study, we present a reliable and tunable 1 T-phase TMDs-based synthetic strategy to atomically engineer non-metal doping sites onto metallic 1 T-WS2. Benefiting from engineering phosphorus atoms, the optimized electrocatalyst shows great enhancement in the HER. We reveal that P doped WS2 will produce 1 T-phase enriched WS2|P catalyst, which is beneficial to expose more active sites and improve the conductivity of the material. More interestingly, the HER activities of 1 T-WS2|P-5 electrocatalyst outperforms 2H-WS2, demonstrating a low overpotential (ηHER = 125 mV @ 10 mA cm−2), a small Tafel slope (HER = 73.73 mV dec−1), and significant durability under acidic conditions. Simultaneously, the 1 T-WS2|P-5 catalyst also presents an overpotential of 190 mV at 10 mA cm−2 and a low Tafel slope of 92.11 mV dec−1 under alkaline conditions. Moreover, the catalyst has excellent catalytic stability under acidic and alkaline conditions. We introduce the first example of 1 T phase enrichment P doped WS2 as a HER electrochemical catalyst, and this work is expected to open a new door for the discovery of other 1 T-phase TMDs as effective catalysts for renewable energy.

Metal-organic frameworks (MOFs), as 3D porous precursors, have demonstrated great potential as electrode materials for lithium storage on account of high specific area and tunable structure. To enhance the intrinsic performance and endow MOFs with versatility, constructing distinctive morphology of MOFs while maintaining their porous structure to satisfy various application is highly desirable. Herein, we successfully developed a facile strategy for rationally designing and synthesizing the morphology of MOFs from circular plate shape to truncated octahedron gem shape via regulating the ratio of Pluronic F127 to terephthalic acid. In addition, the as-prepared Pluronic F127-regulated Ti-based MOFs (F-MIL-125-Ti) sacrifice as precursor to derive 3D mesoporous titanium dioxide (TiO2) and as template for in situ growth of molybdenum disulfide (MoS2) nanosheets to prepare a truncated octahedron TiO2@C@MoS2 hierarchical composite. As expected, the distinctive 3D hierarchical TiO2@C@MoS2 electrode exhibits superb rate capability (775 mAh g−1 at 5 C) and outstanding cycling stability (822 mAh g−1 after 1000 cycles at 1 C), which is benefiting from the 3D mesoporous TiO2 and synergies between TiO2 and MoS2 nanosheets. This work develops new insight to design novel and controllable morphology MOFs-derived materials for application of energy storage devices.

Considerable interest has been motivated in the study of oil-water separation by developing new materials and technologies. Here, we demonstrated a simple and effective method to separate oil-water mixtures using the prepared stainless steel meshes by vapor deposition of liquid state polydimethylsiloxane (PDMS). By changing the heating temperature, underoil superhydrophobic stainless steel meshes and underwater superoleophobic stainless steel meshes can be alternatively achieved. The former is used for heavy oil-water mixtures separation after oil prewetting, while the latter is utilized to separate light oil-water mixtures after water prewetting. As a result, a variety of light or heavy oil-water mixtures could be separated with the efficiency above 99.0%. The separation efficiency maintains high even after 60 separation cycles. In addition, these two types of meshes still take effects under severe environmental situation when they are employed to separate oils from strong acidic, alkaline, salt aqueous solutions, and hot or ice water.

A connected graph G is said to be k-connected if it has more than k vertices and remains connected whenever fewer than k vertices are removed. Feng et al. (2017) presented sufficient conditions based on spectral radius for a graph to be k-connected, k-edge-connected, k-Hamiltonian, k-edge-Hamiltonian, β-deficient and k-path-coverable. In this paper, we present some further sufficient conditions for a graph to be k-connected in terms of signless Laplacian spectral radius, distance spectral radius, distance signless Laplacian spectral radius of G and Wiener index, Harary index of its complement.

Let Γ = ( G , σ ) be a signed graph and A ( Γ ) be its adjacency matrix, where G is the underlying graph of Γ. The rank r ( Γ ) of Γ is the rank of A ( Γ ) . We know that for a signed graph Γ = ( G , σ ) , Γ is balanced if and only if Γ = ( G , σ ) ∼ ( G , + ) . That is, when Γ is balanced, then r ( Γ ) = r ( G ) , where r ( G ) is the rank of the underlying graph G of Γ. A natural problem is that: how about the relations between the rank of an unbalanced signed graph and the rank of its underlying graph? In this paper, we first prove that r ( G ) − 2 d ( G ) ≤ r ( Γ ) ≤ r ( G ) + 2 d ( G ) for an unbalanced signed graph with d ( G ) ≥ 1 , where d ( G ) = | E ( G ) | − | V ( G ) | + ω ( G ) is the dimension of cycle spaces of G, ω ( G ) is the number of connected components of G. As an application, we also prove that 1 − d ( G ) r ( Γ ) r ( G ) ≤ 1 + d ( G ) for an unbalanced signed graph with d ( G ) ≥ 1 . All corresponding extremal graphs are characterized.

The Wiener-type invariants of a simple connected graph G = (V(G),E(G)) can be expressed in terms of the quantities Wf = ?{u,v}?V(G) f(dG(u,v)) for various choices of the function f (x), where dG(u,v) is the distance between vertices u and v in G. In this paper, we mainly give some sufficient conditions for a connected graph to be k-connected, ?-deficient, k-hamiltonian, k-edge-hamiltonian, k-path-coverable or satisfy ?(G)? k.

Let $G^{\sigma}$ be an oriented graph and $S(G^{\sigma})$ be its skew-adjacency matrix, where $G$ is called the underlying graph of $G^{\sigma}$. The skew-rank of $G^{\sigma}$, denoted by $sr(G^{\sigma})$, is the rank of $S(G^{\sigma})$. Denote by $d(G)=|E(G)|-|V(G)|+\theta(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $\theta(G)$ are the edge number, vertex number and the number of connected components of $G$, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76--86] proved that $sr(G^{\sigma})\leq r(G)+2d(G)$ for an oriented graph $G^{\sigma}$, where $r(G)$ is the rank of the adjacency matrix of $G$, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of $sr(G^{\sigma})$ of an oriented graph $G^{\sigma}$ in terms of $r(G)$ and $d(G)$ of its underlying graph $G$ is left open till now. In this paper, we prove that $sr(G^{\sigma})\geq r(G)-2d(G)$ for an oriented graph $G^{\sigma}$ and characterize the graphs whose skew-rank attain the lower bound., Comment: 12 pages

Fe-based electrocatalysts have attracted significant attention on account of the possibility for the realization of low-cost, high efficiency, and stable oxygen evolution reaction (OER) in alkaline solution. In this study, we present a new type of Fe(SexS1−x)2 nanorod bundles, which were successfully synthesized via a simple one-step hydrothermal method. We further demonstrate the applicability of these structures as electrocatalysts for OER application. We reveal that the incorporation of S into FeSe2 cause lattice strain and lattice defect, which is beneficial to expose more active sites and optimize the electronic structure, and hence the Fe(SexS1−x)2 composites display favorable catalytic for OER. More interestingly, the as-prepared Fe(Se0.5S0.5)2 catalyst exhibits the best OER performance with an overpotential at 10 mA·cm−2 of 247 mV, a low tafel slope of 54 mV/dec, and excellent catalytic stability. This work is expected to open a new door to discover other Fe-based materials as efficient catalysts for renewable energy.

Let M be a mixed graph and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H(M)$\end{document}H(M) be its Hermitian-adjacency matrix. If we add a Randić weight to every edge and arc in M, then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randić matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{H}(M)=(r_{h})_{kl}$\end{document}RH(M)=(rh)kl of a mixed graph M, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(r_{h})_{kl}=-(r_{h})_{lk}=\frac{\mathbf{i}}{\sqrt {d_{k}d_{l}}}$\end{document}(rh)kl=−(rh)lk=idkdl (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf{i}=\sqrt{-1}$\end{document}i=−1) if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(v_{k},v_{l})$\end{document}(vk,vl) is an arc of M, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(r_{h})_{kl}=(r_{h})_{lk}=\frac{1}{\sqrt{d_{k}d_{l}}}$\end{document}(rh)kl=(rh)lk=1dkdl if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v_{k}v_{l}$\end{document}vkvl is an undirected edge of M, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(r_{h})_{kl}=0$\end{document}(rh)kl=0 otherwise. In this paper, firstly, we compute the characteristic polynomial of the Hermitian-Randić matrix of a mixed graph. Furthermore, we give bounds on the Hermitian-Randić energy of a general mixed graph. Finally, we give some results about the Hermitian-Randić energy of mixed trees.

Some drawbacks such as lack of durability, requirement for high-cost fluorosilane modification, and complicated fabrication processes have limited the practical applications of superhydrophobic coatings for several years. In this work, we provide a novel method to fabricate a robust superhydrophobic coating without pre-modification by pressing ZnSn(OH)6 (ZHS) particles into polydimethylsiloxane (PDMS) solution, and then curing. PDMS used in this work is not only a low surface energy material, but also an adhesive to enhance the bonding force between particles and substrates. The PDMS@ZHS coating (ZHS particles coated on PDMS) is also superhydrophobic when it is immersed into dodecane, and shows excellent self-cleaning in air and oil environments. After undergoing serious chemical corrosion and mechanical abrasion, the superhydrophobicity of the coating remained well. For different kinds of substrates, this coating can be covered and repels water as well. Owing to the excellent flame-retardancy of ZHS particles, PDMS@ZHS coated paper displays excellent flame retardancy in limiting oxygen index (LOI), vertical flame, and cone calorimeter tests. We believe that this simple, environmentally friendly, and versatile fabrication of the PDMS@ZHS coating has excellent real-life applications.

We study graph theory. The graphs we consider in the thesis consist of a finite vertex set and a finite edge set, and each edge joins an unordered pair of (not necessarily distinct) vertices. In a graph, a round trip which visits a subset of the vertices once, is called a cycle. If such a round trip passes through every vertex of the graph precisely once, we call it a Hamilton cycle. The Hamilton problem, which is the problem of determining whether a Hamilton cycle exists in a given graph has been proved to be NP-complete. So most researchers are focussing on establishing sufficient conditions, i.e., conditions on the graph that guarantee the existence of a Hamilton cycle. In this thesis, we focus on giving sufficient conditions for hamiltonian properties of a graph in terms of structural or algebraic parameters. Besides the Hamilton cycle, we also consider traceability. It requires that there exists a Hamilton path in the graph, i.e., a path through the vertices and edges of the graph that visits all the vertices exactly once (and does not return to the starting vertex). We say a graph is traceable if it contains a Hamilton path, and traceable from a vertex x if it contains a Hamilton x-path, i.e., a Hamilton path starting at vertex x. So, the next hamiltonian property we deal with is that the graph is traceable from every arbitrary vertex. The last hamiltonian property we take into account is Hamilton-connectivity, which requires that any two distinct vertices can be connected by a Hamilton path. The results in this thesis mainly involve degree conditions, spectral conditions, and Wiener index and Harary index conditions for traceability, hamiltonicity or Hamilton-connectivity of graphs. All of our results involve the characterization of the exceptional graphs, i.e., all the nonhamiltonian graphs that satisfy the condition. Our contributions extend existing results.

In this paper, we establish some sufficient conditions for a graph to be Hamilton-connected in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph. Furthermore, we also give some sufficient conditions for a graph to be traceable from every vertex in terms of the edge number, the spectral radius and the signless Laplacian spectral radius.

Hypothesis Surface energy gradient would generate an imbalance force to drive tiny water droplets in dry air from the hydrophilic bumps to superhydrophobic domains, which has found on the Stenocara beetle’s back. Experiments Inspired by this phenomenon, we introduced a pristine superhydrophilic filter paper on the lower surface energy superhydrophobic filter paper. ZnSn(OH)6 particles and polydimethylsiloxane were mixed to prepare the superhydrophobic coating, and the coating was spray-coated on the poly(dialkyldimethylammonium chloride) covered filter paper to separate the span 80 stabilized water-in-isooctane emulsion. A pristine filter paper was added on the superhydrophobic filter paper to fabricate another membrane for separation. Findings The results revealed that with a pristine filter paper, the membrane performed higher efficiency and more recyclability, and it could separate the emulsions with higher surfactant concentrations. The stabilized water droplets passed the superamphiphilic surface, and hindered by the superhydrophobic surface, generating a surface energy gradient for better separation. In addition, the superhydrophobic membrane could be protected from fire to some degree due to the introduced ZnSn(OH)6 particles with excellent flame retardancy. This easy and efficient approach via simply bringing in pristine superhydrophilic membrane has great potential applications for water-in-oil emulsion separation or oil purification.

Let G? be an oriented graph and S(G?) be its skew-adjacency matrix. The skew-rank of G?, denoted by sr(G?), is the rank of S(G?). In this paper, we characterize all the bicyclic oriented graphs with skew-rank 6. Let G? be a bicyclic oriented graph with pendant vertices but no pendant twins. If s r ( G ? ) = 6 , then 6 ? |V(G?)| ? 10.

Let $\Phi=(G, \varphi)$ be a complex unit gain graph (or $\mathbb{T}$-gain graph) and $A(\Phi)$ be its adjacency matrix, where $G$ is called the underlying graph of $\Phi$. The rank of $\Phi$, denoted by $r(\Phi)$, is the rank of $A(\Phi)$. Denote by $\theta(G)=|E(G)|-|V(G)|+\omega(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $\omega(G)$ are the number of edges, the number of vertices and the number of connected components of $G$, respectively. In this paper, we investigate bounds for $r(\Phi)$ in terms of $r(G)$, that is, $r(G)-2\theta(G)\leq r(\Phi)\leq r(G)+2\theta(G)$, where $r(G)$ is the rank of $G$. As an application, we also prove that $1-\theta(G)\leq\frac{r(\Phi)}{r(G)}\leq1+\theta(G)$. All corresponding extremal graphs are characterized., Comment: 17 pages. arXiv admin note: text overlap with arXiv:1612.05043

In this paper, first, we establish a sufficient condition for a bipartite graph to be Hamilton-connected. Furthermore, we also give two sufficient conditions on distance signless Laplacian spectral radius for a graph to be Hamilton-connected and traceable from every vertex, respectively. Last, we obtain a sufficient condition for a graph to be Hamiltonian in terms of the distance signless Laplacian spectral radius of $G^{C}$., Comment: 7 pages, 2 tables, submitted