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.

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Let G= (V, E) be a connected simple graph. A labeling f: V®Z2 induces an edge labeling f*: E®Z2 defined by f* (xy) =f (x)+f (y) for each xy Î E. For i Î Z2, let vf (i) =|f-1 (i)| and ef (i) =|f*-1 (i)|. A labeling f is called FRIENDLY if |vf (1) - vf (0)|£1. The full FRIENDLY INDEX SET of G consists all possible dierences between the number of edges labeled by 1 and the number of edges labeled by 0. In recent years, full FRIENDLY INDEX SETs for certain graphs were studied, such as tori, grids P2´Pn, and cylinders Cm´Pn for some n and m. In this paper we study the full FRIENDLY INDEX SETs of cylinder graphs Cm´P2 for m ³ 3, Cm´P3 for m ³4 and C3´Pn for n ³ 4. The results in this paper complement the existing results in literature, so the full FRIENDLY INDEX SET of cylinder graphs are completely determined.

BACKGROUND AND OBJECTIVES: Cattle farming is an important pillar of dry climate agriculture. Cattle farming has even become the main engine of the community's economy. Currently, there are significant obstacles in the implementation of ecological, economic, technological, and infrastructural dimensions, alongside legal and institutional frameworks, that are essential for promoting sustainable livestock farming practices. Specific studies on different dimensional attributes affecting the sustainability of beef cattle are required. The objective of this study is to investigate different factors within four dimensions that affect the sustainability of cattle farming practices in dry climatic conditions.METHODS: This study was conducted using an interview approach to respondents using a questionnaire or list of questions. The questionnaire contains questions on 37 dimensional attributes, including ten environmental, ten economic, ten technological and infrastructure attributes, and seven legal and institutional attributes. A range of responses was collected from individuals engaged in livestock management in the dry climatic conditions of Timor Tengah Utara Regency. The data that has been collected is analyzed employing Multidimensional Scaling and expedited appraisal software. To evaluate sustainability and leverage attributes, Monte Carlo analysis and the alternating least squares algorithm are applied.FINDINGS: The study's outcomes revealed that the sustainability INDEX pertaining to beef cattle farming in dry climates recorded an average score of 53.09 percent. This finding is based on the economic dimension of 45.58 percent, the environmental dimension of 54.18 percent, the technology and infrastructure dimension of 35.15 percent, and the legal and institutional dimension of 77.46 percent. The sustainability INDEX, reflecting the technology and infrastructure aspects, ranks the lowest, primarily because the technological advancements necessary for achieving sustainable livestock farming are still underdeveloped. The availability of livestock waste processing technology is still relatively low and has not been processed optimally. This situation has an impact on the environment.CONCLUSION: The sustainability INDEX of dry climate beef cattle farming is greatly influenced by the attributes of technology and infrastructure dimensions. The enhancement of sustainability in the future necessitates careful consideration of specific factors, including the accessibility of waste and product processing technologies, advancements in information technology, and the development of road infrastructure. Attributes in the legal and institutional dimensions are the main factors supporting beef cattle's sustainability. Optimizing extension services by increasing the number of livestock extension workers is crucial in accelerating the dissemination of eco-FRIENDLY livestock technoloBeef Cattle; Dry climate; Environmentally FRIENDLY; Multidimensional scaling (MDS); Sustainability assessmenty and innovation.

In recent years, due to the upward growth of the elderly population in the world, this issue has become one of the focus points of researchers. The issue of optimizing social urban structures is the most basic issue in studies related to the elderly in the field of urban planning. In this research, the indicators of measuring the infrastructure of the elderly-FRIENDLY city in the metropolis of Mashhad have been examined. The research method is descriptive-analytical, which was implemented using the survey method. According to the structure of the questionnaires, the statistical population of the research consisted of experts in the urban area, health and hygiene services, psychological services and transportation and information technology in Mashhad city. The sample size was determined using Cochran's formula of 100 people. The research samples were selected using a simple random method. The research data was collected using the standard questionnaire of the World Health Organization. According to the results of the research, Mashhad city is not in a good condition for the elderly in the indicators of housing, outdoor spaces and buildings, community support and health services, civic participation and employment. Also, according to Friedman's test, the average rating of open space and buildings INDEX in the city of Dostdar Salamand has the highest priority, and the communication and information INDEX has the lowest priority among the indicators. The achievement of this research is the precise recognition of the shortcomings of Mashhad participation and employment. metropolis compared to the standard scales of the elderly-FRIENDLY city. New high-yielding and quick-yielding solutions that have not been used in this metropolis so far have been presented to raise the standard level of housing indicators, outdoor spaces and participation and employment have been provided.

The purpose of this paper is to present an INDEX for evaluation of power network security after terrorist attacks against the network facilities using fuzzy SET theory. In order to achieve this goal, a method based on probability and possibility theory is used. In the first step, by using Bayesian network which is a method based on probability theory, the probability of the attack as well as the risk value caused by terrorist activities against the power network facilities was calculated and the network facilities were ranked according to the level of the risk. In the second step, by employing fuzzy load flow and finding the fuzzy parameters related to network performance, the impact of the loss of each component ranked in the first step on power network was investigated. Finally, having calculated the risk value caused by terrorist attacks against the network facilities, an INDEX was provided for evaluating power network security. Also, in order to validate the effectiveness of the proposed method, security INDEX for a specific network was calculated.

Let G be a graph with vertex SET V (G) and edge SET E(G), a vertex labeling f: V (G)! Z2 induces an edge labeling f+: E(G)! Z2 defined by f+(xy) = f(x) + f(y), for each edge xy 2 E(G). For each i 2 Z2, let vf (i) = |{u 2 V (G): f(u) = i}| and ef+(i) = |{xy 2 E(G): f+(xy) = i}|. A vertex labeling f of a graph G is said to be FRIENDLY if |vf (1) − vf (0)|  1. The FRIENDLY INDEX SET of the graph G, denoted by FI(G), is defined as {|ef+(1) − ef+(0)|: the vertex labeling f is FRIENDLY}. The full FRIENDLY INDEX SET of the graph G, denoted by FFI(G), is defined as {ef+(1) − ef+(0): the vertex labeling f is FRIENDLY}. A graph G is cordial if − 1, 0 or 1 2 FFI(G). In this paper, by introducing labeling subgraph embeddings method, we determine the cordiality of a family of cubic graphs which are double-edge blow-up of P2 × Pn, n  2. Consequently, we completely determined FRIENDLY INDEX and full product cordial INDEX SETs of this family of graphs.