An independent set in a graph G is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial AΣx^(|A|), where the sum is over all independent sets A of G. In 1987, Alavi,Malde, Schwe...An independent set in a graph G is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial AΣx^(|A|), where the sum is over all independent sets A of G. In 1987, Alavi,Malde, Schwenk and Erd os conjectured that the independence polynomial of any tree or forest is unimodal.Although this unimodality conjecture has attracted many researchers’ attention, it is still open. Recently, Basit and Galvin even asked a much stronger question whether the independence polynomial of every tree is ordered log-concave. Note that if a polynomial has only negative real zeros then it is ordered log-concave and unimodal.In this paper, we observe real-rootedness of independence polynomials of rooted products of graphs. We find some trees whose rooted product preserves real-rootedness of independence polynomials. In consequence, starting from any graph whose independence polynomial has only real zeros, we can obtain an infinite family of graphs whose independence polynomials have only real zeros. In particular, applying it to trees or forests, we obtain that their independence polynomials are unimodal and ordered log-concave.展开更多
Finding optimal knots is a challenging problem in spline fitting due to a lack of prior knowledge regarding optimal knots.The unimodality of initial B-spline approximations associated with given data is a promising ch...Finding optimal knots is a challenging problem in spline fitting due to a lack of prior knowledge regarding optimal knots.The unimodality of initial B-spline approximations associated with given data is a promising characteristic of locating optimal knots and has been applied successfully.The initial B-spline approximations herein are required to approximate given data well enough and characterized by the unimodality if jumps from the highest-order derivatives of the approximations at some interior knots are local maxima.In this paper,we prove the unimodality of the initial B-spline approximations that are constructed under two assumptions:Data points are sampled uniformly and sufficiently from B-spline functions,and initial knots are chosen as the parameters of sampling points.Our work establishes the theoretical basis of the unimodality of initial B-spline approximations and pioneers the theoretical study of locating optimal knots.展开更多
An independent set in a graph G is a set of pairwise non-adjacent vertices.Let ik(G)denote the number of independent sets of cardinality k in G.Then its generating function I(G;x)=∑^(α(G))^(k=0)ik(G)x^(k),is called ...An independent set in a graph G is a set of pairwise non-adjacent vertices.Let ik(G)denote the number of independent sets of cardinality k in G.Then its generating function I(G;x)=∑^(α(G))^(k=0)ik(G)x^(k),is called the independence polynomial of G(Gutman and Harary,1983).In this paper,we introduce a new graph operation called the cycle cover product and formulate its independence polynomial.We also give a criterion for formulating the independence polynomial of a graph.Based on the exact formulas,we prove some results on symmetry,unimodality,reality of zeros of independence polynomials of some graphs.As applications,we give some new constructions for graphs having symmetric and unimodal independence polynomials.展开更多
In this paper, we consider some properties for bounded linear operators concerning distributional chaos. Norm-unimodality of bounded linear operators im- plies distributional chaos. Some properties such as similarity ...In this paper, we consider some properties for bounded linear operators concerning distributional chaos. Norm-unimodality of bounded linear operators im- plies distributional chaos. Some properties such as similarity and spectra description for norm-unimodal operators are considered. The existence of distributional chaos in nest algebra is also proved. In addition, we obtain a sufficient and necessary condition of distributional chaos for a class of operators, which contains unilateral backward weighted shift operators.展开更多
基金supported by the National Natural Science Foundation of China (Nos.11971206, 12022105)the Natural Science Foundation of Distinguished Young Scholars of Jiangsu Province (No.BK20200048)Postgraduate Research Practice&Innovation Program of Jiangsu Province (No.KYCX21-2565)。
文摘An independent set in a graph G is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial AΣx^(|A|), where the sum is over all independent sets A of G. In 1987, Alavi,Malde, Schwenk and Erd os conjectured that the independence polynomial of any tree or forest is unimodal.Although this unimodality conjecture has attracted many researchers’ attention, it is still open. Recently, Basit and Galvin even asked a much stronger question whether the independence polynomial of every tree is ordered log-concave. Note that if a polynomial has only negative real zeros then it is ordered log-concave and unimodal.In this paper, we observe real-rootedness of independence polynomials of rooted products of graphs. We find some trees whose rooted product preserves real-rootedness of independence polynomials. In consequence, starting from any graph whose independence polynomial has only real zeros, we can obtain an infinite family of graphs whose independence polynomials have only real zeros. In particular, applying it to trees or forests, we obtain that their independence polynomials are unimodal and ordered log-concave.
基金supported by the National Natural Science Foundation of China(No.11801393)the Natural Science Foundation of Jiangsu Province(No.BK20180831).
文摘Finding optimal knots is a challenging problem in spline fitting due to a lack of prior knowledge regarding optimal knots.The unimodality of initial B-spline approximations associated with given data is a promising characteristic of locating optimal knots and has been applied successfully.The initial B-spline approximations herein are required to approximate given data well enough and characterized by the unimodality if jumps from the highest-order derivatives of the approximations at some interior knots are local maxima.In this paper,we prove the unimodality of the initial B-spline approximations that are constructed under two assumptions:Data points are sampled uniformly and sufficiently from B-spline functions,and initial knots are chosen as the parameters of sampling points.Our work establishes the theoretical basis of the unimodality of initial B-spline approximations and pioneers the theoretical study of locating optimal knots.
基金Supported by National Natural Science Foundation of China(Grant Nos.11971206,12022105)Natural Science Foundation for Distinguished Young Scholars of Jiangsu Province(Grant No.BK20200048)。
文摘An independent set in a graph G is a set of pairwise non-adjacent vertices.Let ik(G)denote the number of independent sets of cardinality k in G.Then its generating function I(G;x)=∑^(α(G))^(k=0)ik(G)x^(k),is called the independence polynomial of G(Gutman and Harary,1983).In this paper,we introduce a new graph operation called the cycle cover product and formulate its independence polynomial.We also give a criterion for formulating the independence polynomial of a graph.Based on the exact formulas,we prove some results on symmetry,unimodality,reality of zeros of independence polynomials of some graphs.As applications,we give some new constructions for graphs having symmetric and unimodal independence polynomials.
基金The Youth Foundation of Department of Mathematics,Jilin University
文摘In this paper, we consider some properties for bounded linear operators concerning distributional chaos. Norm-unimodality of bounded linear operators im- plies distributional chaos. Some properties such as similarity and spectra description for norm-unimodal operators are considered. The existence of distributional chaos in nest algebra is also proved. In addition, we obtain a sufficient and necessary condition of distributional chaos for a class of operators, which contains unilateral backward weighted shift operators.
