Let G = (V, E) be a simple graph. An edge LABELING f: E®{0, 1} induces a vertex LABELING f+: V ® Z2 defined by f+ (v) º S uvÎE f (uv) (mod 2) for each v Î V , where Z2 = {0, 1} is the additive group of order 2. For i Î {0, 1}, let ef (i) = ÷f-1 (i)÷ and vf (i) = ÷(f+) -1(i)÷. A LABELING f is called edge-FRIENDLY if ÷ef (1) - ef (0)÷ £ 1. If (G) = vf (1) - vf (0) is called the edge-FRIENDLY index of G under an edge-FRIENDLY LABELING f. Extreme values of edge-FRIENDLY index of complete bipartite graphs will be determined.

Please click on PDF to view the abstract.

Please click on PDF to view the abstract

A difference LABELING of a graph G is an injective function f: V (G) → N ∪ {0} together with the weight function f∗ on E(G) given by f∗(uv) = |f(u)-f(v)| for every edge uv in G. The collection of subgraphs induced by the edges of the same weight is a decomposition of G and is called the common weight decomposition of G induced by f. Let ϒf denote the collection of all the paths taken from each member of the common weight decomposition induced by f. A difference LABELING f of G is said to be a graphoidal difference LABELING if ϒf is an acyclic graphoidal decomposition of G. This paper initiates a study on this concepts.

Let G = (V,E) be a connected simple graph. A LABELING f: ® Z2 induces two edge LABELINGs f+, f*: E® Z2 defined by f+(xy) = f(x)+f(y) and f*(xy) = f(x)f(y) for each xyÎE. For iÎZ2, let vf (i)= |f-1(i)|, e f+ (i)= |(f+)-1(i)| and e f* (i) = |(f*)-1(i)|. A LABELING f is called FRIENDLY if |vf (1)-vf (0)| £1. For a FRIENDLY LABELING f of a graph G, the FRIENDLY index of G under f is defined by if+ (G) = e f+(1) – e f+(0). The set {if+ (G) | f is a FRIENDLY LABELING of G} is called the full FRIENDLY index set of G. Also, the product-cordial index of G under f is defined by if*(G) = ef* (1) – ef* (0). The set {if* (G) | f is a FRIENDLY LABELING of G} is called the full product-cordial index set of G. In this paper, we find a relation between the FRIENDLY index and the product-cordial index of a regular graph. As applications, we will determine the full product-cordial index sets of torus graphs which was asked by Kwong, Lee and Ng in 2010, and those of cycles.

As a part of the human society, children have formed the future generation of every country, and the child's interaction with the surrounding environment requires his physical and mental development. Therefore, it is important to pay attention to the environment that promotes mental and physical development of the child. In order to check the suitable environment for children, the initiative of the child-FRIENDLY city has been taken into consideration. The general purpose of this research is to improve the status of two cities from the perspective of a child-FRIENDLY city and to identify the appropriate indicators of a child-FRIENDLY city and to make a comparative comparison in the two metropolises of Tehran and Isfahan. The current research is developmental in terms of the combined method and in terms of the goal, and it is based on library and documentary studies, field surveys and interviews. Yeoman-Whitney test was used to analyze the indicators. The results of the evaluation show that in the comparison of the two cities of Tehran and Isfahan in terms of the examined indicators (social and physical, economic, environmental, urban management and leisure), Tehran city in the variable of safety and protection with an average of 3.09 and Isfahan city in Participation and citizenship variables with an average of 3.22 have a favorable level. Although Tehran is in a better condition compared to Isfahan in other indicators, in general, both cities are far apart in terms of standard indicators, and suggestions have been made in this regard.

Relation between religion and sociological theories is the most important concern of scholars who study in the field of social sciences.Different views have been expressed in this regard. To draw them on a line one must put on one side those views which reject the existence of any relation between them. On the other side those who believe that faith can take part in this area. Islam as a complete religion has solutions both for spiritual guidance of man and for man’s social life. We should just discover these solutions. New theories such as sociological theories guide us to discover these laws.The present paper lies at the middle of this line between two sides of it. The author has a middle approach. He starts his research with a little question in the field of sociology namely social-LABELING theory and then explores different aspects of this theory and highlights its most important points and finally makes a comparison between them and hadith sources trying to find an answer.