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

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

Energy Consumption Rate based Stable Election Protocol (ECRSEP) for WSNs

In recent few yearsWireless Sensor Networks (WSNs) have seen an increased interest in various applications like border field security, disaster management and medical applications. So large number of sensor nodes are deployed for such applications, which can work autonomously. Due to small power batteries in WSNs, efficient utilization of battery power is an important factor. Clustering is an efficient technique to extend life time of sensor networks by reducing the energy consumption. In this paper, we propose a new protocol; Energy Consumption Rate based Stable Election Protocol (ECRSEP). Our CH selection scheme is based on the weighted election probabilities of each node according to the Energy Consumption Rate (ECR) of each node. We compare results of our proposed protocol with Low Energy Adaptive Clustering Hierarchy (LEACH), Distributed Energy Efficient Clustering (DEEC), Stable Election Protocol (SEP), and Enhanced SEP(ESEP). Our simulation results show that our proposed protocol, ECRSEP outperforms all these protocols in terms of network stability and network lifetime

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

Q-LEACH: A New Routing Protocol for WSNs

Wireless Sensor Networks (WSNs) with their dynamic applications gained a tremendous attention of researchers. Constant monitoring of critical situations attracted researchers to utilize WSNs at vast platforms. The main focus in WSNs is to enhance network life-time as much as one could, for efficient and optimal utilization of resources. Different approaches based upon clustering are proposed for optimum functionality. Network life-time is always related with energy of sensor nodes deployed at remote areas for constant and fault tolerant monitoring. In this work, we propose Quadrature-LEACH (Q-LEACH) for homogenous networks which enhances stability period, network life-time and throughput quiet significantly

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