On integral graphs which belong to the class\overline {\alpha K_{a,a} \cup \beta {\rm K}_{b,b} }

LetG be a simple graph and letG denote its complement. We say thatG is integral if its spectrum consists of integral values. In this work we establish a characterization of integral graphs which belong to the class $\overline {\alpha K_{a,a} \cup \beta {\rm K}_{b,b} } $ wheremG denotes them-fold union of the graphG.     

On Integral Graphs Which Belong to the Class

Let G be a simple graph and let ¯G denote its complement. A graph G is said to be integral if its spectrum contains integral values. If is integral with >1 and a>b, where mG denotes the m-fold union of the graph G, we show that it belongs to the class of integral graphs where (i) (kn ² s ²+km nst+ms,km ² t ²+kmnst+ms)= such that (m,)=1 and (s,)=1; (ii) k,m,n,s,t,z such that (k,m)=1, (k,s)=1, (m,n)=1, (m,s)=1, (n,t)=1 and (s,t)=1; and (iii) ns>m t. Acknowledgments.The author is very grateful to the referees for their valuable comments and suggestions concerning this paper.1991 Mathematics Subject Classification. 05 C 50     

On integral graphs which belong to the class\overline {\alpha K_a \cup \beta K_b }

LetG be a simple graph and let $\bar G$ denotes its complement. We say thatG is integral if its spectrum consists entirely of integers. If $\overline {\alpha K_a \cup \beta K_b } $ is integral we show that it belongs to the class of integral graphs $$\overline {[\frac{{kt}}{\tau }x_o + \frac{{mt}}{\tau }z]K(t + \ell n)k + \ell m \cup [\frac{{kt}}{\tau }y_o + \frac{{(t + \ell n)k + \ell m}}{\tau }z]nK\ell m,} $$ where (i) t, k, l, m, n ∈ ? such that (m, n) =1, (n, t) =1 and (l, t)=1; (ii) τ=((t+ln)k+lm, mt) such that τ| kt; (iii) (x₀, y₀) is aparticular solution of the linear Diophantine equation ((t+ln)k+lm)x-(mt)y=τ and (iv) z≥z₀ where z₀ is the least integer such that $(\frac{{kt}}{\tau }x_0 + \frac{{mt}}{\tau }z_0 ) \geqslant 1$ and $(\frac{{kt}}{\tau }y_0 + \frac{{(t + \ell n)k + \ell m}}{\tau }z_0 ) \geqslant 1$ .     

第二大特征值小于$\frac{\sqrt{5}-1}{2}$的平面图

若能将图$G$画在一个平面上,使得任何两条边仅在顶点处相交,则称$G$是平面图.本文刻画了第二大特征值小于$\frac{\sqrt{5}-1}{2}$的所有无孤立点的平面图.     

完全解决图$\overline{B_{n-8,1,4}}$的色等价类

Two graphs are defined to be adjointly equivalent if and only if their complements are chromatically equivalent.Using the properties of the adjoint polynomials and the fourth character R₄（G）,the adjoint equivalence class of graph B_n-8,1,4 is determined.According to the relations between adjoint polynomial and chromatic polynomial,we also simultaneously determine the chromatic equivalence class of B_n-8,1,4 that is the complement of B_n-8,1,4.     

Integral geometry for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {D}}$\end{document}‐modules on dual flag manifolds and generalized Verma modules

Given a pair of dual generalized flag manifolds of a semisimple algebraic group, we show that the integral transform between them given by the open orbit in their product is an equivalence. We also describe the links of this problem with the structure of generalized Verma modules, and how the above construction can be applied to the representation theory of real forms of the group.     

图类aKa,a\βCP(b)中的整谱图

设图G是一个简单图,图G的补图记为G,如果G的谱都是整数.就称G是整谱图.鸡尾酒会图CP(n)=K2n-nK2(K2n是2n阶完全图)和完全二部图K…都是整谱图.确定了图类 aKa,a∪βCP中的所有的整谱图.     

Application of Almost Increasing Sequence for Absolute Riesz $|\overline{N}, p_n^{\alpha, \beta}; \delta|_k$ Summable Factor

In this paper, we generate an extended result by Bor and Seyhan concerning absolute Riesz summability factors. Further, we develop some well-known results from our main result.     

On the tangential Cauchy‐Fueter operators on nondegenerate quadratic hypersurfaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {H}^2}$\end{document}

On quadratic hypersurfaces in $\mathbb {H}^2$, we find the explicit forms of tangential Cauchy‐Fueter operators and associated tangential Laplacians □_b. Then by using the Fourier transformation on the associated nilpotent Lie groups of step two, we construct the relative fundamental solutions to the tangential Laplacians and Szegö kernels on the nondegenerate quadratic hypersurfaces. It is different from the complex case that the quaternionic tangential structures on the nondegenerate quadratic hypersurfaces in $\mathbb {H}^2$ cannot be reduced to one standard model and the non‐homogeneous tangential Cauchy‐Fueter equations are solvable even in many convex cases.     

On the Dynamics of x_{\bf{n} + \bf{1}} = {{ \mgreek{a} + \mgreek{g} x_{n-1} + \mgreek{d} x_{n-2}} \over {A + x_{n-2}}}

We investigate the global stability, the periodic character and the boundedness nature of solutions of the equation in the title for all admissible nonnegative values of the parameters and the initial conditions. We show that the solutions exhibit a trichotomy character depending on how the parameter γ compares to the sum of the parameters δ and A.     

$\mathbb{C}^n$中一类星形映射子族的增长定理及推广的Roper-Suffridge算子

在有界星形圆形域上定义了一个新的星形映射子族, 它包含了$\alpha$阶星形映射族和$\alpha$阶强星形映射族作为两个特殊子类. 给出了此类星形映射子族的增长定理和掩盖定理. 另外, 还证明了Reinhardt域$\Omega_{n,p_{2},\cdots,p_{n}}$上此星形映射子族在Roper-Suffridge算子 \begin{align*} F(z)=\Big(f(z_{1}),\Big(\frac{f(z_{1})}{z_{1}}\Big)^{\beta_{2}}(f'(z_{1}))^{\gamma_{2}}z_{2},\cdots, \Big(\frac{f(z_{1})}{z_{1}}\Big)^{\beta_{n}}(f'(z_{1}))^{\gamma_{n}}z_{n}\Big)' \end{align*} 作用下保持不变, 其中 $\Omega_{n,p_{2},\cdots,p_{n}}=\{z\in {\mathbb{C}}^{n}:|z_1|^2+|z_2|^{p_2}+\cdots + |z_n|^{p_n}<1\}$, $p_{j}\geq1$, $\beta_{j}\in$ $[0, 1]$, $\gamma_{j}\in[0, \frac{1}{p_{j}}]$满足$\beta_{j}+\gamma_{j}\leq1$, 所取的单值解析分支使得 $\big({\frac{f(z_{1})}{z_{1}}}\big)^{\beta_{j}}\big|_{z_{1}=0}=1$, $(f'(z_{1}))^{\gamma_{j}}\mid_{{z_{1}=0}}=1$, $j=2,\cdots,n$. 这些结果不仅包含了许多已有的结果, 而且得到了新的结论.     

一类非线性积分方程正解的存在唯一性

本文研究积分方程ｕ（ｘ）＝λ∫_Ωｋ（ｘ，ｙ）ｆ（ｙ，ｕ（ｙ））ｄｙ，λ＞０及其它的非线性摄动ｕ（ｘ）＝λ∫_Ωｋ（ｘ，ｙ）ｆ（ｙ，ｕ（ｙ））ｄｙ＋Ｇ（ｕ（ｘ）），在ｋ（ｘ，ｙ）非负可测，ｆ（ｘ，ｕ），Ｇ（ｕ）满足一定条件下，得到所述方程解的存在唯一性及其迭代逼近．