title: Star Decompositions of Bipartite Graphs

authors: Lim, Nick Jin Sean

subjects: Graph Decomposition

abstract: In Chapter 1, we will introduce the definitions and the notations used throughout this thesis. We will also survey some prior research pertaining to graph decompositions, with special emphasis on star-decompositions and decompositions of bipartite graphs. Here we will also introduce some basic algorithms and lemmas that are used in this thesis.

In Chapter 2, we will focus primarily on decomposition of complete bipartite graphs. We will also cover the necessary and sufficient conditions for the decomposition of complete bipartite graphs minus a 1-factor, also known as crown graphs and show that all complete bipartite graphs and crown graphs have  a decomposition into stars when certain necessary conditions for the decomposition are met. This is an extension of the results given in "On claw-decomposition of complete graphs and complete bigraphs" by Yamamoto, et. al. We will propose a construction for the decomposition of the graphs. 

In Chapter 3, we focus on the decomposition of complete equipartite tripartite graphs. This result is similar to the results of "On Claw-decomposition of complete multipartite graphs" by Ushio and Yamamoto. Our proof is again by construction and we propose how it might extend to equipartite multipartite graphs. We will also discuss the 3-star decomposition of complete tripartite graphs.

In Chapter 4 , we will discuss the star decomposition of 4-regular bipartite graphs, with particular emphasis on the decomposition of 4-regular bipartite graphs into 3-stars. We will propose methods to extend our strategies to model the problem as an optimization problem. We will also look into the probabilistic method discussed in "Tree decomposition of Graphs" by Yuster and how we might modify the results of this paper to star decompositions of bipartite graphs. 

In Chapter 5, we summarize the findings in this thesis, and discuss the future work and research in star decompositions of bipartite and multipartite graphs.

text: Star Decompositions of Bipartite Graphs In Chapter 1, we will introduce the definitions and the notations used throughout this thesis. We will also survey some prior research pertaining to graph decompositions, with special emphasis on star-decompositions and decompositions of bipartite graphs. Here we will also introduce some basic algorithms and lemmas that are used in this thesis.

In Chapter 2, we will focus primarily on decomposition of complete bipartite graphs. We will also cover the necessary and sufficient conditions for the decomposition of complete bipartite graphs minus a 1-factor, also known as crown graphs and show that all complete bipartite graphs and crown graphs have  a decomposition into stars when certain necessary conditions for the decomposition are met. This is an extension of the results given in "On claw-decomposition of complete graphs and complete bigraphs" by Yamamoto, et. al. We will propose a construction for the decomposition of the graphs. 

In Chapter 3, we focus on the decomposition of complete equipartite tripartite graphs. This result is similar to the results of "On Claw-decomposition of complete multipartite graphs" by Ushio and Yamamoto. Our proof is again by construction and we propose how it might extend to equipartite multipartite graphs. We will also discuss the 3-star decomposition of complete tripartite graphs.

In Chapter 4 , we will discuss the star decomposition of 4-regular bipartite graphs, with particular emphasis on the decomposition of 4-regular bipartite graphs into 3-stars. We will propose methods to extend our strategies to model the problem as an optimization problem. We will also look into the probabilistic method discussed in "Tree decomposition of Graphs" by Yuster and how we might modify the results of this paper to star decompositions of bipartite graphs. 

In Chapter 5, we summarize the findings in this thesis, and discuss the future work and research in star decompositions of bipartite and multipartite graphs.

year: 2015

title: Aggregate investment behaviour in New Zealand: An examination of the theory of investment behaviour and its econometric representation

authors: Lichtwark, Philip Laurence

abstract: In this study an attempt is made to construct a model explaining quarterly investment in plant and machinery in the New Zealand economy, with a view to providing a basis for forecasting investment expenditures. A survey of theory and econometric models reveals that in the more successful overseas models the principle determinants of investment expenditures are changes in demand as reflected output changes, the capital stock, which gives an indication of existing capacity and determines replacement requirements, and also factors affecting the cost of capital equipment. These factors form the basis of the theory of investment behaviour. However it is found that the pure theoretical framework, although useful, does not provide a complete explanation of investment in New Zealand, and that such factors as import controls and other exogenous influences affect investment in this country. 
The final model purports to explain ninety three per cent of the variation in investment expenditures and is statistically sound. This was considered a good result in view of the data problems encountered. Although the model is not entirely satisfactory the equation performed reasonably well in ex-poste forecasts which suggests that with some improvements, particularly of some of the data series, it could provide a useful indication of the effects of policy measures on investment.

text: Aggregate investment behaviour in New Zealand: An examination of the theory of investment behaviour and its econometric representation In this study an attempt is made to construct a model explaining quarterly investment in plant and machinery in the New Zealand economy, with a view to providing a basis for forecasting investment expenditures. A survey of theory and econometric models reveals that in the more successful overseas models the principle determinants of investment expenditures are changes in demand as reflected output changes, the capital stock, which gives an indication of existing capacity and determines replacement requirements, and also factors affecting the cost of capital equipment. These factors form the basis of the theory of investment behaviour. However it is found that the pure theoretical framework, although useful, does not provide a complete explanation of investment in New Zealand, and that such factors as import controls and other exogenous influences affect investment in this country. 
The final model purports to explain ninety three per cent of the variation in investment expenditures and is statistically sound. This was considered a good result in view of the data problems encountered. Although the model is not entirely satisfactory the equation performed reasonably well in ex-poste forecasts which suggests that with some improvements, particularly of some of the data series, it could provide a useful indication of the effects of policy measures on investment.

year: 1973

title: He purapura whetū tō te whakataukī

authors: Tuahine, Hōhepa

subjects: Whakataukī

abstract: Ko te whakataukī he kōrero e whakaatu ana i ngā tini āhuatanga o te Māori me te āhua o tōna taiao (Milroy, 2004, wh. 38).  Hei tuinga ake, he kōrero e pupuri ana i ngā tohutohu e tau ai te noho o te tangata ki te whenua.  Ka rerekē te whakatakotoranga o te kupu o te whakataukī o nehe, ki te reo Māori o nāianei.  Ko ngā whakataukī o tāukiuki i titoa i te wā e ora tonu ana ngā whare wānanga o nehe me ōna tohunga.  Ka mutu, ko te reo me ōna kupu kei tērā taumata e rere ana.  Nō reira, he kōrero e whakaata ana i te ao Māori, ko ōna atua, ko ōna tikanga, ko ōna pakanga, ko ōna tātai whakapapa, ko ōna mātauranga, ko ōna tohutohu (Milroy, 2004, wh. 38).  Ka tākaitia ake aua mātauranga ki te kōrero poto noa iho, ka tuku iho ai ki ngā uri whakaheke.

text: He purapura whetū tō te whakataukī Ko te whakataukī he kōrero e whakaatu ana i ngā tini āhuatanga o te Māori me te āhua o tōna taiao (Milroy, 2004, wh. 38).  Hei tuinga ake, he kōrero e pupuri ana i ngā tohutohu e tau ai te noho o te tangata ki te whenua.  Ka rerekē te whakatakotoranga o te kupu o te whakataukī o nehe, ki te reo Māori o nāianei.  Ko ngā whakataukī o tāukiuki i titoa i te wā e ora tonu ana ngā whare wānanga o nehe me ōna tohunga.  Ka mutu, ko te reo me ōna kupu kei tērā taumata e rere ana.  Nō reira, he kōrero e whakaata ana i te ao Māori, ko ōna atua, ko ōna tikanga, ko ōna pakanga, ko ōna tātai whakapapa, ko ōna mātauranga, ko ōna tohutohu (Milroy, 2004, wh. 38).  Ka tākaitia ake aua mātauranga ki te kōrero poto noa iho, ka tuku iho ai ki ngā uri whakaheke.

year: 2016