Journal of Mathematical Extension
Volume:11 Issue: 3, Summer 2017

In a search for triangle-free graphs with arbitrarily large chromaticnumbers, Mycielski developed a graph transformation that transformsa graph G into a new graph (G), we now call the Mycielskian ofG, which has the same clique number as G and whose chromatic numberequals (G)+1. This paper presents exact values of the equitable chromaticnumber = for the Mycielski's graph of complete graphs (Kn),the Mycielski's graph of cycles (Cn), the Mycielski's graph of paths(Pn), the Mycielski's graph of Helm graphs (Hn) and the Mycielski'sgraph of Gear graphs (Gn).

In this paper we give a characterization of fuzzy normed linear space and then two examples are introduced. By using this characterization some fuzzy fixed point theorems are established.

We propose an iterative method that solves an unconstrained nonconvex nonsmooth optimization problem. The proposed method is a descent method that uses subgradients at each iteration and contains very simple procedures for finding descent directions and for solving line search subproblems. The convergence of the algorithms is studied

We introduce the notion of modified Hardy-Rogers type F-contraction on closed ball, F-multiplicative contraction on closed ball and obtain some new fixed point results for such contractions. Some comparative examples are constructed to illustrate these results. The existence of the solution of family of Volterra type integral equations is shown via fixed point methods.

Let g and h be arbitrary elements of a given finite group G. Then g and h are said to be autoconjugate if there exists some automorphism α of G such that h = gα. In this article, we introduce and study auto-average length of autoconjugacy classes of finite groups. Also, we construct some sharp bounds for the auto-average length of finite groups

It is well-known that every group is equal to the direct product of its subgroup and related left and right transversalsets (in the sense of direct product of subsets). Therefore, every subgroup of a group is its left and right factor,and an its consequence is the Lagrange theorem for finite groups.This paper generalizes the results for semigroupsand proves a necessary and sufficient condition for a subgroup of asemigroup to be a factor.Also, by using the conception upper periodic subsets of semigroups and groups (introduced byM.H. Hooshmand as a generalization of the conception ideals) we provesome sufficient conditions for a vast class of subsets of semigroups to be factorsand Lagrange subsets.

Let R be a commutative ring and M be an R-module with a proper submodule N. A generalization of total graphs, denoted by T(ΓN H(M)), is introduced and investigated. It is the (undirected) graph with all elements of M as vertices and for distinct x; y 2 M, the vertices x; y are adjacent if and only if x + y 2 MH(N) where MH(N) = fm 2 M : rm 2 N for some r 2 Hg and H is a multiplicatively closed subset of R. In this paper, in addition to studying some algebraic properties of MH(N), we investigate some graph theoretic properties of two essential subgraphs of T(ΓN H(M)).

This paper deals with the nonlinear dynamics‎, ‎chaos‎, ‎optimal and adaptive control of an epidemic model for H1N1 influenza with unknown parameters‎. ‎Two different control strategies are explored‎. ‎First‎, ‎we use the optimal control theory to reduce the infected individuals and the cost of vaccination‎. ‎Then‎, ‎we study the problem of optimal control of unstable steady-states of H1N1 influenza system using a nonlinear‎ ‎control approach‎. ‎Finally‎, ‎we propose the Lyapunov stability to control of the chaotic epidemic model of influenza with unknown parameters by a feedback control approach‎. ‎Matlab bvp4c and ode45 have been used for solving the autonomous chaotic systems and the extreme conditions obtained from the Pontryagin's maximum principle (PMP)‎. ‎Furthermore‎, ‎numerical simulations are included to demonstrate the effectiveness of the proposed control strategies.