Fixing number of co-noraml product of graphs

An automorphism of a graph $G$ is a bijective mapping from the vertex set of $G$ to itself which preserves the adjacency and the non-adjacency relations of the vertices of $G$. A fixing set $F$ of a graph $G$ is a set of those vertices of $G$ which when assigned distinct labels removes all the automorphisms of $G$, except the trivial one. The fixing number of a graph $G$, denoted by $fix(G)$, is the smallest cardinality of a fixing set of $G$. The co-normal product $G_1\ast G_2$ of two graphs $G_1$ and $G_2$, is a graph having the vertex set $V(G_1)\times V(G_2)$ and two distinct vertices $(g_1, g_2), (\acute{g_1}, \acute{g_2})$ are adjacent if $g_1$ is adjacent to $\acute{g_1}$ in $G_1$ or $g_2$ is adjacent to $\acute{g_2}$ in $G_2$. We define a general co-normal product of $k\geq 2$ graphs which is a natural generalization of the co-normal product of two graphs. In this paper, we discuss automorphisms of the co-normal product of graphs using the automorphisms of its factors and prove results on the cardinality of the automorphism group of the co-normal product of graphs. We prove that $max\{fix(G_1), fix(G_2)\}\leq fix(G_1\ast G_2)$, for any two graphs $G_1$ and $G_2$. We also compute the fixing number of the co-normal product of some families of graphs.Comment: 13 page

Analyzing Energy-efficiency and Route-selection of Multi-level Hierarchal Routing Protocols in WSNs

The advent and development in the field of Wireless Sensor Networks (WSNs) in recent years has seen the growth of extremely small and low-cost sensors that possess sensing, signal processing and wireless communication capabilities. These sensors can be expended at a much lower cost and are capable of detecting conditions such as temperature, sound, security or any other system. A good protocol design should be able to scale well both in energy heterogeneous and homogeneous environment, meet the demands of different application scenarios and guarantee reliability. On this basis, we have compared six different protocols of different scenarios which are presenting their own schemes of energy minimizing, clustering and route selection in order to have more effective communication. This research is motivated to have an insight that which of the under consideration protocols suit well in which application and can be a guide-line for the design of a more robust and efficient protocol. MATLAB simulations are performed to analyze and compare the performance of LEACH, multi-level hierarchal LEACH and multihop LEACH.Comment: NGWMN with 7th IEEE Inter- national Conference on Broadband and Wireless Computing, Communication and Applications (BWCCA 2012), Victoria, Canada, 201

An analysis of the rights of minorities in international law

The concern for the plight of the minorities and an attempt to safeguard their interests has been an ideal which has contributed towards the growth and expansion of international law. Although international law primarily operates through the medium of States, and minorities generally have no locus standi, the treatment which the minorities receive from their States has occasionally become a matter of international concern. International law, however, has historically found it difficult to deal with the issue of minorities. Like the poor, the weak and the inarticulate, they have, since time immemorial been the natural victims of persecution and genocide. In an age when wars were "just", religious repression legitimate, and cultural or political dissidence unacceptable, minorities remained the prime target of repression.Even in this contemporary period of relative tolerance and rationality, minorities are often subjected to persecution, discrimination and genocide. The stance of international law remains tentative and extremely cautious, for minorities pose questions of a serious nature, existing in myriad forms with their own social, political, cultural and religious peculiarities. Often transcending national frontiers minorities are extremely capable of appealing to the sensitivities of their international sympathisers. Most national boundaries are arbitrarily drawn, and a number of States contain turbulent factions artificially placed within their borders, often cutting across frontiers. Many regions continue to witness a perpetual and infinite struggle between the minority groups on the one hand and the State on the other, sometimes to a point where the very fabric of the institution of the State comes under threat.A consideration of many of the contemporary disputes including those involving the Kurds of Iraq, Turkey and Iran, the Kashmiris of Pakistan and India, the Sikhs of Indian Punjab, the Tamils of Sri Lanka, the Biharis of Bangladesh, the Tibetans of China, the Catholics of Northern Ireland, the non-Arab indigenous Africans of Southern Sudan, and protagonists in civil wars of the fonner Yugoslavia and the Soviet Union, reveals the widespread nature of the conflict. While a number of governments attempt to hide behind Article 2(7) of the United Nations Charter and take refuge in the "citadel" of State sovereignty and sovereign equality, the minorities may take to heart the revolutionary indoctrination of secession in the name of self-determination. A significant number of States that emerged from the rubble of decolonisation have, in particular, faced serious challenges from their minorities. The emphasis on the principle of Uti Possidetis, though redolent of the colonial past, meant arbitrary divisions of peoples belonging to the same tribe, race, or religion. In the face of these challenges a number of States attempted to introduce constitutional devices including integration and pluralism, while others, insensitive to minority aspirations and unwilling to compromise, concentrated on the building of a nation-state with one dominant culture, language, politics and religion. Consequences of some of these policies have been severe, resulting in enduring and painful conflict.The issue of minority rights has and continues to occupy a sensitive position in inter-State relations. Historical as well as current events show that the subject is also capable of engulfing the international community as a whole. The ending of the "Cold-war" has brought with it a need for urgent revision of many areas of international law and international relations. The subject of the rights of minorities in international law, it is submitted, is worthy of an extremely thorough reconsideration

Cross-sectional study identifying forms of tobacco used by Shisha smokers in Pakistan

OBJECTIVES: To estimate the frequency of different forms of tobacco intake such as smoker\u27s tobacco, chewable tobacco and snuff tobacco among shisha smoker\u27s and to study the patterns and predictors of shisha smoking affecting youth from different cities of Pakistan. METHODS: A cross-sectional study was conducted including youth from four cities. Participants were asked to fill out a data collection tool at shisha cafes, shopping malls and restaurants. Data was analyzed using SPSSv.18. RESULTS: A total of 406 participants, 296 (73%) males and 110 (27%) females were included in the study. There were 163 (40%) cigarette smokers; 65 (16%) chewed tobacco and 33 (8%) snuffed it. The median age at initiation of Shisha smoking was 20 years. 280 (69%) considered Shisha smoking to be less deleterious to health than cigarettes. Respiratory disease was the most commonly cited health effect reported. Most 248 (61%) of the participants were infrequent shisha smokers. CONCLUSION: There is high frequency of tobacco usage in the form of cigarettes, chewable tobacco and snuff tobacco among shisha smokers of Pakistan. The highest frequency is for cigarette smoking. The rise in Shisha smoking as a trendy social habit appears to be occurring despite emerging scientific evidence of its potential health risks

Navigating the Digital Transformation: The Impact of E-Banking Channels on Bank Performance in Pakistan\u27s Evolving Financial Landscape

E-banking is a game changer that revolutionized the financial landscape by breaking the barriers of 9 to 5 traditional physical banking. COVID-19 signifies its importance. This study examines the dynamic relationship between various e-banking channels in Pakistan and bank performance from 2006 to 2022 while controlling macroeconomic variables like GDP and inflation. Results confirmed that past performance is dominantly affecting current performance. Customer perception toward e-banking channels also shapes its relationship with bank performance. ATMs are perceived as secure, easy, and can be used 24/7 across the country, which is why the desired positive relationship is confirmed with the bank\u27s performance. However, due to high operational and maintenance costs, security risks, and lack of customer adoption, the desired positive impact of mobile banking and e-commerce has not been realized. Banks also need to focus on customer education to improve their financial literacy, maintenance costs, operational costs, secured infrastructure, and efficient governance to protect the customers\u27 information and hard-earned money. This will increase customer confidence in using these channels despite the associated risks

Principals Perception Regarding Factors Affecting The Performance Of Teachers

This study investigated the perception of principals on how the factors of subject mastery, teaching methodology, personal characteristics, and attitude toward students affect the performance of teachers at higher secondary level in the Punjab. All principals of higher secondary level in the Punjab were part of the population of the study. From the population, 120 principals were selected as the sample. A questionnaire was developed and validated through pilot testing. The data obtained were tabulated and analyzed by using statistical techniques of mean and standard deviation. The major conclusions of the study were that the factor of subject mastery was perceived to be influencing the performance of teachers maximally, but the factor of attitude toward students was affecting the performance of teachers minimally. The remaining two factors - teaching methodology and personal characteristics - were perceived to be at the intermediary level

Analytical Survey of Wearable Sensors

Wearable sensors inWireless Body Area Networks (WBANs) provide health and physical activity monitoring. Modern communication systems have extended this monitoring remotely. In this survey, various types of wearable sensors discussed, their medical applications like ECG, EEG, blood pressure, detection of blood glucose level, pulse rate, respiration rate and non medical applications like daily exercise monitoring and motion detection of different body parts. Different types of noise removing filters also discussed at the end that are helpful in to remove noise from ECG signals. Main purpose of this survey is to provide a platform for researchers in wearable sensors for WBANs.Comment: BioSPAN with 7th IEEE International Conference on Broadband and Wireless Computing, Communication and Applications (BWCCA 2012), Victoria, Canada, 201