We determine the minimum size of n-factor-critical graphs and that of k-extendable bipartite graphs, by considering Harary graphs and related graphs. Moreover, we determine the minimum size of k-extendable non-bipartite graphs for k = 1, 2, and pose a related conjecture for general k.

هدف: این پژوهش با هدف بررسی اثر آموزش برنامه هوش موفق استرنبرگ بر افزایش خلاقیت و تحمل ابهام دانش‌­آموزان انجام شد. روش: روش ­آزمایشی طرح پیش ­آزمون پس پیگیری گروه کنترل بود. جامعۀ آماری را 15842 نفر ازدانش‌­آموزان پسر کلاس چهارم در سال تحصیلی1399-1400 شهر اصفهان تشکیل دادند که30 نمونه‌­گیری تصادفی چند مرحله­‌ای انتخاب به صورت دو آزمایش (15 نفر) گمارده شدند. ده جلسه برنامه‌­ی دریافت نمود. داده­...

A.M. Ostrowski in 1951 gave two well-known upper bounds for the spectral radius of nonnegative matrices. However, the bounds are not of much practical use because they all involve a parameter α in the interval [0, 1], and it is not easy to decide the optimum value of α. In this paper, their equivalent forms which can be computed with the entries of matrix and without having to minimize the expr...

We study an impartial game introduced by Anderson and Harary. This game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nimnumbers of this game for generalized dihedral groups, which are of the form Dih(A) = Z2 n A for a finite abelian group A.

Since Ostrowski introduced the class of M{matrices more than fty equivalent deenitions of these matrices were established. Here we give another characterization using Schur complements.

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We introduce a modification of the Harary index where the contributions of vertex pairs are weighted by the sum of their degrees. After establishing basic mathematical properties of the new invariant, we proceed by finding the extremal graphs and investigating its behavior under several standard graph products.

We present new functional equations for the species of plane and of planar (in the sense of Harary and Palmer, 1973) 2-trees and some associated pointed species. We then deduce the explicit molecular expansion of these species, i.e. a classification of their structures according to their stabilizers. There result explicit formulas in terms of Catalan numbers for their associated generating seri...

In this paper we establish a strong relationship between the set of lazy representations and the set of paths in a Sturmian graph associated with a real number α. We prove that for any non-negative integer i the unique path weighted i in the Sturmian graph associated with α represents the lazy representation of i in the Ostrowski numeration system associated with α. Moreover, we provide several...

In this paper we first generalize the Ostrowski inequality on time scales for k points and then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on time scales as special cases.