An extension of Maclaurins inequality
نویسنده
چکیده
Let G be a graph of order n and clique number !: For every x = (x1; : : : ; xn) 2 Rn and 1 s !; set fs (G;x) = X fxi1 : : : xis : fi1; : : : ; isg is an s-clique of Gg ; and let s (G;x) = fs (G;x) ! s 1 : We show that if x 0; then 1 (G;x) 1=2 2 (G;x) 1=! ! (G;x) : This extends the inequality of Maclaurin (G = Kn) and generalizes the inequality of Motzkin and Straus. In addition, if x > 0; for every 1 s < ! we determine when 1=s s (G;x) = 1=(s+1) s+1 (G;x). Letting ks (G) be the number of s-cliques of G; we show that the above inequality is equivalent to the combinatorial inequality k1 (G) ! 1 k2 (G) ! 2 !1=2 k! (G) ! ! !1=! : These results complete and extend earlier results of Motzkin and Straus, Khadzhiivanov, Fisher and Ryan, and Petingi and Rodriguez. AMS classi cation: Keywords: Maclaurins inequality, clique number, number of cliques Our graph-theoretic notation follows [1]; in particular, all graphs are de ned on the vertex set f1; 2; : : : ; ng = [n] and G (n) stands for a graph with n vertices. We write ! (G) for the size of the maximal clique of G; Ks (G) for the set of s-cliques of G; and ks (G) for jKs (G)j. For any graph G = G (n) ; vector x = (x1; : : : ; xn) 2 R; and 1 s ! = ! (G) ; set fs (G;x) = X fxi1 : : : xis : fi1; : : : ; isg 2 Ks (G)g
منابع مشابه
Extension of Hardy Inequality on Weighted Sequence Spaces
Let and be a sequence with non-negative entries. If , denote by the infimum of those satisfying the following inequality: whenever . The purpose of this paper is to give an upper bound for the norm of operator T on weighted sequence spaces d(w,p) and lp(w) and also e(w,?). We considered this problem for certain matrix operators such as Norlund, Weighted mean, Ceasaro and Copson ma...
متن کاملOn a $k$-extension of the Nielsen's $beta$-Function
Motivated by the $k$-digamma function, we introduce a $k$-extension of the Nielsen's $beta$-function, and further study some properties and inequalities of the new function.
متن کاملAn Extension of a Geometric Inequality of Finite Point Set on a Sphere in the Constant Curvature Space
In this paper, we first prove an algebraic inequality, then use it obtain an extension of a geometirc inequality in the n -dimensional constant curvature space.
متن کاملOperator Extensions of Hua’s Inequality
Abstract. We give an extension of Hua’s inequality in pre-Hilbert C∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is eq...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007