The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. We say a graph G is star extremal if its circular chromatic number is equal to its...The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. We say a graph G is star extremal if its circular chromatic number is equal to its fractional chromatic number. This paper gives an improvement of a theorem. And we show that several classes of circulant graphs are star extremal.展开更多
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 tw...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 ).展开更多
The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. A graph is called star extremal if its fractional chromatic number equals to its c...The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. A graph is called star extremal if its fractional chromatic number equals to its circular chromatic number (also known as the star chromatic number). This paper studies the star extremality of the circulant graphs whose generating sets are of the form {±1,±k} .展开更多
In this paper, it is shown that： if λ1 ,……λs axe nonzero real numbers, not all of the same sign, such that A1/A2 is irrational, then for any real number η and ε 〉 0 the inequality ｜λ1x1^2 ＋ λ2x2^2＋ λ3x3^...In this paper, it is shown that： if λ1 ,……λs axe nonzero real numbers, not all of the same sign, such that A1/A2 is irrational, then for any real number η and ε 〉 0 the inequality ｜λ1x1^2 ＋ λ2x2^2＋ λ3x3^4＋ λsx3^4＋……λsx8^4 ＋η〈 ε has infinitely many solutions in positive integers x1,... ,xs.展开更多
The main methods of the second phase quantitative analysis in current material science researches are manual recognition and extracting by using software such as Image Tool and Nano Measurer. The weaknesses such as hi...The main methods of the second phase quantitative analysis in current material science researches are manual recognition and extracting by using software such as Image Tool and Nano Measurer. The weaknesses such as high labor intensity and low accuracy statistic results exist in these methods. In order to overcome the shortcomings of the current methods, the Ω phase in A1-Cu-Mg-Ag alloy is taken as the research object and an algorithm based on the digital image processing and pattern recognition is proposed and implemented to do the A1 alloy TEM （transmission electron microscope） digital images process and recognize and extract the information of the second phase in the result image automatically. The top-hat transformation of the mathematical morphology, as well as several imaging processing technologies has been used in the proposed algorithm. Thereinto, top-hat transformation is used for elimination of asymmetric illumination and doing Multi-layer filtering to segment Ω phase in the TEM image. The testing results are satisfied, which indicate that the Ω phase with unclear boundary or small size can be recognized by using this method. The omission of these two kinds of Ω phase can be avoided or significantly reduced. More Ω phases would be recognized （growing rate minimum to 2% and maximum to 400% in samples）, accuracy of recognition and statistics results would be greatly improved by using this method. And the manual error can be eliminated. The procedure recognizing and making quantitative analysis of information in this method is automatically completed by the software. It can process one image, including recognition and quantitative analysis in 30 min, but the manual method such as using Image Tool or Nano Measurer need 2 h or more. The labor intensity is effectively reduced and the working efficiency is greatly improved.展开更多
Let G be a graph with degree sequence ( dv). If the maximum degree of any subgraph induced by a neighborhood of G is at most m, then the independence number of G is at least $\sum\limits_v {f_{m + 1} \left( {d_v } \ri...Let G be a graph with degree sequence ( dv). If the maximum degree of any subgraph induced by a neighborhood of G is at most m, then the independence number of G is at least $\sum\limits_v {f_{m + 1} \left( {d_v } \right)} $ , where fm+1( x) is a function greater than $\frac{{log\left( {x/\left( {m + 1} \right)} \right) - 1}}{x}for x > 0$ for x> 0. For a weighted graph G = ( V, E, w), we prove that its weighted independence number (the maximum sum of the weights of an independent set in G) is at least $\sum\limits_v {\frac{{w_v }}{{1 + d_v }}} $ where wv is the weight of v.展开更多
文摘The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. We say a graph G is star extremal if its circular chromatic number is equal to its fractional chromatic number. This paper gives an improvement of a theorem. And we show that several classes of circulant graphs are star extremal.
文摘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 ).
文摘The circular chromatic number and the fractional chromatic number are two generalizations of the ordinary chromatic number of a graph. A graph is called star extremal if its fractional chromatic number equals to its circular chromatic number (also known as the star chromatic number). This paper studies the star extremality of the circulant graphs whose generating sets are of the form {±1,±k} .
基金the National Natural Science Foundation of China(10671056)
文摘In this paper, it is shown that： if λ1 ,……λs axe nonzero real numbers, not all of the same sign, such that A1/A2 is irrational, then for any real number η and ε 〉 0 the inequality ｜λ1x1^2 ＋ λ2x2^2＋ λ3x3^4＋ λsx3^4＋……λsx8^4 ＋η〈 ε has infinitely many solutions in positive integers x1,... ,xs.
基金Project(51171209)supported by the National Natural Science Foundation of China
文摘The main methods of the second phase quantitative analysis in current material science researches are manual recognition and extracting by using software such as Image Tool and Nano Measurer. The weaknesses such as high labor intensity and low accuracy statistic results exist in these methods. In order to overcome the shortcomings of the current methods, the Ω phase in A1-Cu-Mg-Ag alloy is taken as the research object and an algorithm based on the digital image processing and pattern recognition is proposed and implemented to do the A1 alloy TEM （transmission electron microscope） digital images process and recognize and extract the information of the second phase in the result image automatically. The top-hat transformation of the mathematical morphology, as well as several imaging processing technologies has been used in the proposed algorithm. Thereinto, top-hat transformation is used for elimination of asymmetric illumination and doing Multi-layer filtering to segment Ω phase in the TEM image. The testing results are satisfied, which indicate that the Ω phase with unclear boundary or small size can be recognized by using this method. The omission of these two kinds of Ω phase can be avoided or significantly reduced. More Ω phases would be recognized （growing rate minimum to 2% and maximum to 400% in samples）, accuracy of recognition and statistics results would be greatly improved by using this method. And the manual error can be eliminated. The procedure recognizing and making quantitative analysis of information in this method is automatically completed by the software. It can process one image, including recognition and quantitative analysis in 30 min, but the manual method such as using Image Tool or Nano Measurer need 2 h or more. The labor intensity is effectively reduced and the working efficiency is greatly improved.
基金Li is supported in part by the National Natural Science Foundation of China (Grant No. 19871023) Science Foundation of the Education Ministry of China "333" Foundation and NSF of Jiangsu Province. Zang is supported in part by an RGC earmarked researc
文摘Let G be a graph with degree sequence ( dv). If the maximum degree of any subgraph induced by a neighborhood of G is at most m, then the independence number of G is at least $\sum\limits_v {f_{m + 1} \left( {d_v } \right)} $ , where fm+1( x) is a function greater than $\frac{{log\left( {x/\left( {m + 1} \right)} \right) - 1}}{x}for x > 0$ for x> 0. For a weighted graph G = ( V, E, w), we prove that its weighted independence number (the maximum sum of the weights of an independent set in G) is at least $\sum\limits_v {\frac{{w_v }}{{1 + d_v }}} $ where wv is the weight of v.
