Geodetic Number and Geo-Chromatic Number of 2-Cartesian Product of Some Graphs

A set S ⊆ V (G) is called a geodetic set if every vertex of G lies on a shortest u-v path for some u, v ∈ S, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by . A set C ⊆ V (G) is called a chromatic set if C contains all vertices of different colors in G, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by . A geo-chromatic set Sc ⊆ V (G) is both a geodetic set and a chromatic set. The geo-chromatic number of G is the minimum cardinality among all geo-chromatic sets of G. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.

The Factorization of Adjoint Polynomials of E^G（i）-class Graphs and Chromatically Equivalence Analysis

Let Sn be the star with n vertices, and let G be any connected graph with p vertices. We denote by Eτp＋（r-1）^G（i） the graph obtained from Sr and rG by coinciding the i-th vertex of G with the vertex of degree r - 1 of S,, while the i-th vertex of each component of （r - 1）G be adjacented to r - 1 vertices of degree 1 of St, respectively. By applying the properties of adjoint polynomials, We prove that factorization theorem of adjoint polynomials of kinds of graphs Eτp＋（r-1）^G（i）∪（r - 1）K1 （1 ≤i≤p）. Furthermore, we obtain structure characteristics of chromatically equivalent graphs of their complements.

Circular Chromatic Numbers of Some Distance Graphs

The circular chromatic number of a graph is an important parameter of a graph. The distance graph G(Z,D) , with a distance set D , is the infinite graph with vertex set Z={0,±1,±2,...} in which two vertices x and y are adjacent iff y-x∈D . This paper determines the circular chromatic numbers of two classes of distance graphs G(Z,D m,k,k+1 ) and G(Z,D m,k,k+1,k+2 ).

Review:Chromatic Dispersion Manipulation Based on Optical Metasurfaces

Metasurfaces are densely arrayed two⁃dimensional(2D)artificial planar metamaterials,which can manipulate the polarization,distribution,and amplitude of light by accurately controlling the phase of the scattering light.The flat metasurface has the potential to substantially reduce the thickness and complexity of the structures and allows ease of fabrication and integration into devices.However,the inherent chromatic aberration of the metasurface originating from the resonant dispersion of the antennas and the intrinsic chromatic dispersion limit their quality.How to effectively suppress or manipulate the chromatic aberration of metalenses has attracted worldwide attention in the last few years,leading to a variety of excellent achievements.Furthermore,utilizing the chromatic dispersion of metasurface to realize special functionalities is also of significant importance.In this review,the most promising recent examples of chromatic dispersion manipulation based on optical metasurface materials are highlighted and put into perspective.

CIRCULAR CHROMATIC NUMBER AND MYCIELSKI GRAPHS

For a general graph G, M（G） denotes its Mycielski graph. This article gives a number of new sufficient conditions for G to have the circular chromatic number xc（M（G）） equals to the chromatic number x（M（G））, which have improved some best sufficient conditions published up to date.

EDGE-FACE CHROMATIC NUMBER OF 2-CONNECTED PLANE GRAPHS WITH HIGH MAXIMUM DEGREE

The edge-face chromatic number Xef （G） of a plane graph G is the least number of colors assigned to the edges and faces such that every adjacent or incident pair of them receives different colors. In this article, the authors prove that every 2-connected plane graph G with △（G）≥｜G｜ - 2≥9 has Xef（G） = △（G）.

Graph Description by the Chromatic Polynomials

By means of the chromatic polynomials, this paper provided a necessary and sufficient condition for the graph G being a mono-cycle graph（the Theorem 1）, a first class hi-cycle graph and a second class bicycle graph（the Theorem 2）, respectively.

Estimation of illumination chromaticity via adaptive reduced relevance vector machine

A new regression algorithm of an adaptive reduced relevance vector machine is proposed to estimate the illumination chromaticity of an image for the purpose of color constancy. Within the framework of sparse Bayesian learning, the algorithm extends the relevance vector machine by combining global and local kernels adaptively in the form of multiple kernels, and the improved locality preserving projection （LLP） is then applied to reduce the column dimension of the multiple kernel input matrix to achieve less training time. To estimate the illumination chromaticity, the algorithm is trained by fuzzy central values of chromaticity histograms of a set of images and the corresponding illuminants. Experiments with real images indicate that the proposed algorithm performs better than the support vector machine and the relevance vector machine while requiring less training time than the relevance vector machine.

Chromatic Adaptation to Complex Field

Chromatic adaptation of human eyes to complex field is investigated using binocular asymmetric matching method. The results show that the leve of chromatic adapta- tion mainly depends on the average color of adapting field, whereas no obvious change oc- curs when the number of hues and their color differences vary in a complex field. The shifts of perceived color evoked by chromatic adaptation in CIE 1976 L U V color space are approximately proportional to the average color difference between the adapting field and reference field.

Effect of Varieties on Tea Pigment and Chromatic Aberration of Black Tea Infusion

Effects of varieties on the content of tea pigment, L＊, a＊ and b＊ value of chromatic aberration of black tea infusion was studied. The correlation relationship between content of TF, TR, TB and L＊, a＊, b＊ value was extremely significant at P＆lt;0.05. Fresh tea leaves with plucking standard of 1 bud and 2 leaves could man-ufacture black tea with higher content of tea pigment, higher a＊ and b＊ value but lower L＊ value, than those of plucking standard of 1 bud and 3 leaves. But the discrepancy of certain parameter of the two plucking standards changed with vari-eties. Varieties’ quality potential could be concealed by improper plucking standards. Six varieties were divided into four groups on the basis of gross tea polyphenols in fresh tea leaves （GTP）, consumption of tea polyphenols to gross tea polyphenols ratio （CTP/GTP） and gross tea pigment to consumption of tea polyphenols ratio （GP/CTP）. Hybrid performed the very suitability of black tea manufacturing. To en-sure high quality of made black tea, suitable variety should be chosen and reason-able plucking standard also should be taken into consideration.

Chromatic and achromatic differences of melanin-and carotenoid-based plumage coloration in five minivet species(Pericrocotus spp.)under conspecific and predator visual systems

Adaptive mate choice has been accepted as the leading theory to explain the colorful plumage of birds.This theory hypothesizes that conspicuous colors act as signals to advertise the qualities of the owners.However,a dilemma arises in that conspicuous colors may not only attract mates,but also alert predators.The"private channels of communication"hypothesis proposes that some intraspecific signals may not be visible to heterospecific animals because of different visual systems.To better understand the evolution of plumage colors and sexual selection in birds,here we studied the chromatic difference and achromatic differences of melanin-and carotenoid-based plumage coloration in five minivet species(Pericrocotus spp.)under conspecific and predator visual systems.We found that either the chromatic or achromatic difference among male or female minivets’plumage was consistently higher under conspecific vision than under predator vision for all five studied species of minivets.This result indicated that individual differences in plumage colors of minivets were visible to the conspecific receivers and hidden from potential predators as a result of evolution under predation risk and conspecific communication.However,males were under a higher risk of predation because they were more conspicuous than females to the vision of a nocturnal predator.

CLASSIFICATION OF COMPLETE 5-PARTITE GRAPHS AND CHROMATICITY OF 5-PARTITE GRAPHS WITH 5n VERTICES

For a graph G,P(G,λ)denotes the chromatic polynomial of G.Two graphs G and H are said to be chromatically equivalent,denoted by G～H,if P(G,λ)=p(H,λ).Let [G]={H|H～G}.If [G]={G},then G is said to be chromatically unique.For a complete 5 partite graph G with 5n vertices, define θ(G)=(α(G,6)-2 n+1 -2 n-1 + 5)/2 n-2 ,where α(G,6) denotes the number of 6 independent partition s of G.In this paper, the authors show that θ(G)≥0 and determine all g raphs with θ(G)=0,1,2,5/2,7/2,4,17/4.By using these results the chromaticity of 5 partite graphs of the form G-S with θ(G)=0,1,2,5/2,7/2,4,17/4 is inve stigated,where S is a set of edges of G.Many new chromatically unique 5 partite graphs are obtained.

An Approach to Chromaticity of K_4-homeomorphs

We introduce a new method for studying the chromaticity of K4-homeomorphs,by which some new results are obtained.

On Chromatically Equivalence of a Class of Graphs

A class of new graphs is defined. A sufficient condition for pairs of these graphs to be chromatically equivalent is proven.

The Chromatic Uniqueness of Bipartite Graphs K(m,n)-A with |A|=2

The chromatically uniqueness of bipartite graphs K （m, n） - A（]A] = 2） was studied. With comparing the numbers of partitions into r color classes of two chromatically equivalent graphs, one general numerical condition guaranteeing that K（ m, n） - A （ I A ] = 2） is chromatically unique were obtained. This covers and improves the former correlative results.

The Further Results of the Chromatic Uniqueness of Certain Bipartite Graphs K(m, n)-A

With its comprehensive application in network information engineering （e. g. dynamic spectrum allocation under different distance constraints ） and in network combination optimization （e. g. safe storage of deleterious materials）, the graphs＇ cloring theory and chromatic uniqueness theory have been the forward position of graph theory research. The later concerns the equivalent classification of graphs with their color polynomials and the determination of uniqueness of some equivalent classification under isomorphism. In this paper, by introducing the concept of chromatic normality and comparing the number of partitions of two chromatically equivalent graphs, a general numerical condition guarenteeing that bipartite graphs K （ m, n） - A （A belong to E（K （m, n） ） and ｜ A ｜≥ 2） is chromatically unique was obtained and a lot of chromatic uniqueness graphs of bipartite graphs K（m, n） - A were determined. The results obtained in this paper were general. And the results cover and extend the majority of the relevant results obtained within the world.

Some results on circular chromatic number of a graph

For two integers k and d with （k, d） = 1 and k≥2d, let G^dk be the graph with vertex set {0,1,…k - 1 } in which ij is an edge if and only if d≤｜ i -j I｜≤k - d. The circular chromatic number χc（G） of a graph G is the minimum of k/d for which G admits a homomorphism to G^dk. The relationship between χc（ G- v） and χc （G）is investigated. In particular, the circular chromatic number of G^dk - v for any vertex v is determined. Some graphs withx χc（G - v） =χc（G） - 1 for any vertex v and with certain properties are presented. Some lower bounds for the circular chromatic number of a graph are studied, and a necessary and sufficient condition under which the circular chromatic number of a graph attains the lower bound χ- 1 ＋ 1/α is proved, where χ is the chromatic number of G and a is its independence number.

The Circular Chromatic Number of Some Special Graphs

The circular chromatic number of a graph is a natural generalization of the chromatic number. Circular chromatic number contains more information about the structure of a graph than chromatic number does. In this paper we obtain the circular chromatic numbers of special graphs such as C t k and C t k-v, and give a simple proof of the circular chromatic number of H m,n .

Chromatic processing for feature extraction of PD-induced UHF signals in GIS

Partial discharge(PD) detection is an effective means of discovering insulation faults in gas-insulated switchgear(GIS). One of the most extensively used methods in PD detection has historically been the ultrahigh frequency(UHF) method. This study evaluates the chromatic processing methodology and its key factors for feature extraction of UHF signals in GIS. Three types of artificial defects are installed in the GIS tank at 0°, 90°, and 180°, respectively. The features of the UHF signals are extracted in the chromatic space, and PD discrimination of the defects is achieved. The influences of processors are studied before the feature selections are suggested. The time-stepping method is proposed to determine the rules of UHF signal frequency characteristics that vary with time. Finally, the process and options of the chromatics-inspired methodology are summarized.

Novel colored flames via chromaticity of essential colors

Colored flame compositions have distinctive variety of applications ranging from military signaling,rocket tracking, and illuminating devices. Certain elements and compounds when heated to high temperature are able to emit unique wavelengths in the visible region. This study, reports on the development of novel colored flames that cannot be generated by emitting atomic/molecular species. This was achieved by using chromaticity of basic colored flames. Mixing of high quality primary colored flames including Blue, Yellow, and Red in proper ratio was conducted; any interfering incandescent emission resulted from MgO was eliminated using Al metal fuel. The spectral characteristics in terms of luminous intensity, and color quality were evaluated using digital luxmeter and UV-Vis. spectrometer respectively.High quality mixed colored flames include violet, sweet pink, and marigold were developed. This study shaded the light on the state of the art for the real development of novel colored flame compositions and chromaticity of basic colored flames.