Edge Uv Research Articles

Signed graphs with cut points whose positive inertia indexes are two

A signed graph Gσ consists of an underlying graph G and a sign function σ, which assigns each edge uv of G a sign σ(uv), either positive or negative. The adjacency matrix of Gσ is defined as A(Gσ)=(au,vσ) with au,vσ=σ(uv)au,v, where au,v=1 if u,v∈V(G) are adjacent, and au,v=0 otherwise. The positive inertia index of Gσ, written as p(Gσ), is defined to be the number of positive eigenvalues of A(Gσ). Recently, Yu et al. (2016) [12] characterized the signed graphs Gσ with pendant vertices such that p(Gσ)=2. In this paper, we extend the above work to a more general case, characterizing the signed graphs Gσ with cut points whose positive inertia index is 2.

Optical, structural properties and antibacterial activities of uncapped and HMT capped ZnO nanoparticles

In this work, uncapped and HMT (hexamethylenetetramine or hexamine) capped zinc oxide nanoparticles (NPs) have been synthesized by solvothermal method followed by annealing at 200, 400 and 500°C temperatures for 2h in order to tune their antibacterial properties. Effect of annealing temperature on structural growth of ZnO nanostructure have been extensively investigated and reported the synthesis of beautiful cauliflower-like ZnO nanostructures by simply controlling the temperature. HMT capped ZnO NPs have shown enhanced antibacterial activity as compared to that of uncapped one. Changes in optical and structural properties have also been found to be occurred because of capping and thermal annealing. From the X-ray diffraction measurements it has been observed that the synthesized ZnO NPs have hexagonal wurtzite phase. Various calculations through deformation models have been used to investigate the particle sizes, stress, strain and energy density values. Moreover, calculated mean particle size of the prepared NPs using these deformation models are highly correlated with transmission electron micrograph image analyses and Williamson-Hall analyses. Strong near band edge UV photoluminescence emission has also been observed from all the samples. The synthesized ZnO nanomaterials can be used in light emitting devices as well as in antibacterial activities.

Effect of Thickness on Micro-Structural and Optical Properties of Al-Doped ZnO Films Prepared by Sol-Gel Spin Coating

Aluminium (Al) doped Zinc oxide (ZnO) thin films of different thicknesses were prepared on glass substrates by sol-gel spin coating method. The effect of thicknesses on micro-structural and optical properties was investigated for transparent conducting oxide (TCO) application in solar cells and other optoelectronic applications. Grazing incidence x-ray diffraction (GIXRD) showed maximum orientation along (002) plane of c-axis. The variation of different structural parameters like crystallite size, micro-strain, c-axis strain, dislocation density as a function of film thickness was investigated. The FTIR spectra confirmed the formation of Al-doped ZnO film. FESEM images showed spherical shaped nanosized grains and formation of micro pores. The optical absorption increased and absorption peak shifted towards longer wavelength (red shift) with increase in the thickness of the film respectively. The optical transmittance of all the films has a transparency of more than 75% in the visible region. The optical band gap (Eg) decreased with increase in the film thickness. The diffused reflectance (DRS) showed very low reflectance in the region of 400-800 nm, but increased in the 800-900 nm region. Photoluminescence (PL) spectra of the prepared films showed intense band edge UV and low intense visible emissions respectively. The effect of thickness of Al-doped ZnO film on micro-structure, surface morphology, optical absorption and transmittance, diffused reflectance and PL have been investigated and the results are reported.

Neighbor sum distinguishing colorings of graphs with maximum average degree less than $$\tfrac{{37}} {{12}}$$3712

Let G be a graph and let its maximum degree and maximum average degree be denoted by Δ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ′∑(G). Flandrin et al. proposed the following conjecture that χ′∑ (G) ≤ Δ(G) + 2 for any connected graph with at least 3 vertices and G ≠ C5. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < $$\tfrac{{37}} {{12}}$$ and Δ(G) ≥ 7.

Neighbor sum distinguishing total coloring of planar graphs without 5-cycles

Let G be a graph, a proper total coloring ϕ:V(G)∪E(G)→{1,2,…,k} is called neighbor sum distinguishing if f(u)≠f(v) for each edge uv∈E(G), where f(v)=∑uv∈E(G)ϕ(uv)+ϕ(v), v∈V(G). We use χΣ″(G) to denote the smallest number k in such a coloring of G. Pilśniak and Woźniak have already conjectured that χΣ″(G)≤Δ(G)+3 for any simple graph with maximum degree Δ(G). In this paper, we prove that for any planar graph G without 5-cycles, χΣ″(G)≤max⁡{Δ(G)+3,10}.

Neighbor sum distinguishing total chromatic number of K 4-minor free graph

A k-total coloring of a graph G is a mapping ϕ: V (G) ⋃ E(G) → {1; 2,..., k} such that no two adjacent or incident elements in V (G) ⋃ E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v: We say that ϕ is a k-neighbor sum distinguishing total coloring of G if f(u) 6 ≠ f(v) for each edge uv ∈ E(G): Denote χΣ″(G) the smallest value k in such a coloring of G: Pilśniak and Woźniak conjectured that for any simple graph with maximum degree Δ(G), χΣ″ ≤ Δ(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K4-minor free graph G with Δ(G) > 5; χΣ″ = Δ(G) + 1 if G contains no two adjacent Δ-vertices, otherwise, χΣ″(G) = Δ(G) + 2.

Degree sums for edges and cycle lengths in graphs

Let G be a graph of order n satisfying d(u) + d(v) n for every edge uv of G. We show that the circumference - the length of a longest cycle - of G can be expressed in terms of a certain graph parameter, and can be computed in polynomial time. Moreover, we show that G contains cycles of every length between 3 and the circumference, unless G is complete bipartite. If G is 1-tough then it is pancyclic or G = Kr,r with r = n/2.

Sequences of radius [formula omitted] for complete bipartite graphs

A k-radius sequence for a graph G is a sequence of vertices of G (typically with repetitions) such that for every edge uv of G vertices u and v appear at least once within distance k in the sequence. The length of a shortest k-radius sequence for G is denoted by fk(G). We give an asymptotically tight estimation on fk(G) for complete bipartite graphs which matches a lower bound, valid for all bipartite graphs. We also show that determining fk(G) for an arbitrary graph G is NP-hard for every constant k>1.

Color-blind index in graphs of very low degree

Let c:E(G)→[k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v)=(a1,…,ak), where ai is the number of edges incident to v with color i. Reorder c̄(v) for every v in G in nonincreasing order to obtain c∗(v), the color-blind partition of v. When c∗ induces a proper vertex coloring, that is, c∗(u)≠c∗(v) for every edge uv in G, we say that c is color-blind distinguishing. The minimum k for which there exists a color-blind distinguishing edge coloring c:E(G)→[k] is the color-blind index of G, denoted dal(G). We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if dal(G)≤2 is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when dal(G) is finite for a class of 3-regular graphs.

Steinberg-like theorems for backbone colouring

A function f:V(G)→{1,…,k} is a (proper) k-colouring of G if ∣f(u)−f(v)∣≥1, for every edge uv∈E(G). The chromatic numberχ(G) is the smallest integer k for which there exists a proper k-colouring of G.Given a graph G and a subgraph H of G, a circular q-backbone k-colouring f of (G,H) is a k-colouring of G such that q≤∣f(u)−f(v)∣≤k−q, for each edge uv∈E(H). The circularq-backbone chromatic number of a graph pair (G,H), denoted CBCq(G,H), is the minimum k such that (G,H) admits a circular q-backbone k-colouring.Inspired by Steinberg’s conjecture, we conjecture that if G is a planar graph containing no cycles on 4 or 5 vertices and H⊆G is a forest, then CBC2(G,H)≤6. In this work, we first show that if G is a planar graph containing no cycle on 4 or 5 vertices and H⊆G is a forest, then CBC2(G,H)≤7. Then, we prove that if H⊆G is a forest whose connected components are paths, then CBC2(G,H)≤6.

Super edge trimagic total labeling of some star type graphs

A bijection such that for each edge uv ϵ E(G), the value of f(u) + f(uv) + f(v) is equal to either k1 or k2 or k3 is called an edge trimagic total labeling of a (p, q) graph. In this paper we prove that the star type graphs admits edge trimagic total labeling and super edge trimagic total labeling.

The neighbour-sum-distinguishing edge-colouring game

Let γ:E(G)⟶N∗=N∖{0} be an edge colouring of a graph G and σγ:V(G)⟶N∗ the vertex colouring given by σγ(v)=∑e∋vγ(e) for every v∈V(G). A neighbour-sum-distinguishing edge-colouring of G is an edge colouring γ such that for every edge uv in G, σγ(u)≠σγ(v). The neighbour-sum-distinguishing edge-colouring game on G is the 2-player game defined as follows. The two players, Alice and Bob, alternately colour an uncoloured edge of G. Alice wins the game if, when all edges are coloured, the so-obtained edge colouring is a neighbour-sum-distinguishing edge-colouring of G. Otherwise, Bob wins.In this paper we study the neighbour-sum-distinguishing edge-colouring game on various classes of graphs. In particular, we prove that Bob wins the game on the complete graph Kn, n≥3, whoever starts the game, except when n=4. In that case, Bob wins the game on K4 if and only if Alice starts the game.

Effect of doping on the surface modification of nebulizer sprayed BaxZn1−xO nanocrystalline thin films

The influence of Ba doped zinc oxide films were investigated by nebulizer spray pyrolysis technique at 673 K. X-ray diffraction reveals the polycrystalline hexagonal (wurtzite) crystal structure with (0 0 2) preferential orientation. Energy dispersive spectroscopy confirms the presence of Ba, Zn and O elements in the films. Field emission scanning electron microscopy shows that the surface morphology of the nanocrystalline films were changed from spherical shape structure to flake net-like shape and sprout like spherical structure with average grain size is ~100 nm due to the critical doping concentration. PL spectra prominent peaks corresponding to near band edge UV emission and intrinsic defect of the visible blue light region and defect related deep level green emission regions were discussed. The films are highly transparent in the visible region with a transmittance higher than 74%, and have an optical band gap energy values are increased from 3.22 eV to 4.02 eV depending on the Ba doping concentration. Interparticle like grains, grain boundary effect of deposited films is studied by complex impedance spectroscopy.

Hitting times for random walks on subdivision and triangulation graphs

Let G be a connected graph. The subdivision graph of G, denoted by S(G), is the graph obtained from G by inserting a new vertex into every edge of G. The triangulation graph of G, denoted by R(G), is the graph obtained from G by adding, for each edge uv, a new vertex whose neighbours are u and v. In this paper, we first provide complete information for the eigenvalues and eigenvectors of the probability transition matrix of a random walk on S(G) (res. R(G)) in terms of those of G. Then we give an explicit formula for the expected hitting time between any two vertices of S(G) (res. R(G)) in terms of those of G. Finally, as applications, we show that, the relations between the resistance distances, the number of spanning trees and the multiplicative degree-Kirchhoff index of S(G) (res. R(G)) and G can all be deduced from our results directly.

Photocatalytic activity of Ag/ZnO core–shell nanoparticles with shell thickness as controlling parameter under green environment

Plasmonic Ag/ZnO core–shell nanoparticles have been synthesized via a simple two-step wet chemical method for application in Photocatalysis. The morphology, size, crystal structure, composition and optical properties of the nanoparticles are investigated by x-ray diffraction, transmission electron microscopy (TEM), FTIR spectroscopy, ultraviolet–visible (UV–Vis) absorption spectroscopy and photoluminescence (PL) spectroscopy. The shell thicknesses are varied by varying the concentration of zinc nitrate hexa-hydrate and triethanolamine. The ZnO shell coating over Ag core enhances the charge separation, whereas the larger shell thickness and increased refractive index of surrounding medium cause red shifts of surface Plasmon resonance (SPR) peak of Ag core. The photoluminescence (PL) spectra of Ag/ZnO core–shell show that the larger shell thickness quenches the near band edge UV emission of ZnO. The electrochemical impedance spectra (EIS) i.e. Nyquist plots also confirm the higher charge transfer efficiency of the Ag/ZnO core–shell nanoparticles. The Photocatalytic activities of Ag/ZnO core–shell nanoparticles are investigated by the degradation of methylene blue (MB) dye under direct sunlight irradiation. Compared to pure ZnO nanoparticles (NPs), Ag/ZnO core–shell NPs display efficient sunlight plasmonic photocatalytic activity because of the influence of SPR of Ag core and the electron sink effect. The photocatalytic activity of Ag/ZnO core–shell NPs is found to be enhanced with increase in shell thickness.

On the signless Laplacian index and radius of graphs

A bag Bagp,q is a graph obtained from a complete graph Kp by replacing an edge uv by a path Pq. In this paper, we show that for all the connected graphs of order n≥5 with signless Laplacian index q1(G) and radius rad(G), q1(G)⋅rad(G) is maximum for and only for the graph Bagn−2s+3,2s−1, where s=⌈n/4⌉. This solves a conjecture in [6] on the signless Laplacian index involving the radius.

Neighbor sum distinguishing edge coloring of subcubic graphs

A proper edge-k-coloring of a graph G is a mapping from E(G) to {1, 2,..., k} such that no two adjacent edges receive the same color. A proper edge-k-coloring of G is called neighbor sum distinguishing if for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different fromthe sumof colors taken on the edges incident to v. Let χ′Σ(G) denote the smallest value k in such a coloring of G. This parameter makes sense for graphs containing no isolated edges (we call such graphs normal). The maximum average degree mad(G) of G is the maximum of the average degrees of its non-empty subgraphs. In this paper, we prove that if G is a normal subcubic graph with mad(G) < 5/2, then χ′Σ(G) ≤ 5. We also prove that if G is a normal subcubic graph with at least two 2-vertices, 6 colors are enough for a neighbor sum distinguishing edge coloring of G, which holds for the list version as well.

Sum divisor cordial labeling for star and ladder related graphs

A sum divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, . . . , |V(G)|} such that an edge uv is assigned the label 1 if 2 divides f (u) + f (v) and 0 otherwise; and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we prove that D2(K1,n), S' (K1,n), D2(Bn,n), DS(Bn,n), S' (Bn,n), S(Bn,n), , S(Ln), Ln O K1, SLn, TLn, TLn O Ki and CHn are sum divisor cordial graphs.

Bar Visibility Numbers for Hypercubes and Outerplanar Digraphs

A t-bar visibility representation of a graph G assigns each vertex a union of at most t horizontal segments ("bars") in the plane so that vertices u and v are adjacent if and only if some bar assigned to u and some bar assigned to v are joined by an unobstructed vertical line of sight having positive width. For an oriented graph G, having an edge uv requires visibility from a bar for u upward to a bar for v. The visibility number of a graph or digraph G, written b(G), is the least t such that G has a t-bar visibility representation. For the n-dimensional hypercube $$Q_n$$Qn, Euler's Formula yields $$b(Q_n)\ge \lceil \frac{n+1}{4}\rceil $$b(Qn)??n+14?. To prove equality, we decompose $$Q_{4k-1}$$Q4k-1 explicitly into k spanning subgraphs whose components have the form [InlineEquation not available: see fulltext.]. When G is an outerplanar oriented graph, always $$b(G)\le 2$$b(G)≤2; we give a forbidden substructure characterization for those with $$b(G)=1$$b(G)=1.

Super geometric mean graphs

Let G be a graph and f:V(G)→{1,2,3,…,p+q} be an injection. For each edge uv, the induced edge labeling f* is defined as f*(uv)=f(u)f(v). Then f is called a super geometric mean labeling if f(V(G))∪​{ f*(uv);uv∈E(G) }={1,2,3,…,p+q}. A graph that admits a super geometric mean labeling is called a super geometric mean graph. In this paper, we discuss the super geometric meanness of union of any paths, union of any cycles of order ≥5, the graph Pn⊙Sm for m≤3, square graph, total graph, the H-graph, the graph G⊙S1 and G⊙S2 for any H-graph G, subdivision of K1,3 and some chain graphs.