The construction industry in Nigeria has continuously witnessed rapid development as a result of massive investments in infrastructural projects such as housing. The continuous growth of this industry and the conventional approach to construction practices in Nigeria have negatively affected the environment and the wellbeing of the populace. Therefore, the concept of green architecture, also known as sustainable architecture, is a new approach in Nigeria’s construction industry that strives to achieve environmental sustainability. However, various factors have hindered its adoption and utilisation. This study, therefore, examined the various factors hindering the use of green architecture through various literature reviewed and administered questionnaires to built environment professionals in Nigeria to ascertain their perception of those identified factors. Data gathered from the questionnaires were analysed using descriptive statistical tools and ranked according to each factor’s mean index score and relative importance index. The results of the study revealed the most prominent factors hindering the utilisation of green architecture in Nigeria. Hence, findings from this study suggest that more efforts such as public enlightenment and the provisions of incentives are needed to be done by the government, built environment professionals, and other stakeholders in Nigeria’s construction industry for the promotion of green architecture.

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

The concept of graph coloring has become a very active field of research that enhances many practical applications and theoretical challenges. Various methods have been applied in carrying out this study. Let G be a finite group. In this paper, we introduce a new graph of groups, which is a commuting order product prime graph of finite groups as a graph having the elements of G as its vertices and two vertices are adjacent if and only if they commute and the product of their order is a prime power. This is an extension of the study for order product prime graph of finite groups. The graph's general presentations on dihedral groups, generalized quaternion groups, quasi-dihedral groups, and cyclic groups have been obtained in this paper. Moreover, the commuting order product prime graph on these groups has been classified as connected, complete, regular, or planar. These results are used in studying various and recently introduced chromatic numbers of graphs.

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The coprime probability and graph have been studied for various groups by many researchers focusing on the generalization of the probability part. For the coprime graph, the types and properties of the graph have been investigated and the patterns that can be found within a group are analysed. The coprime probability of a group is defined as the probability that the order of a random pair of elements in the group are relatively prime or coprime. Meanwhile, the coprime graph can be explained as a graph whose vertices are elements of a group and two distinct vertices are adjacent if and only if the greatest common divisor of the order of the first vertex and order of the second vertex is equal to one. It was unfortunate that the exploration of probabilities and graphs of groups have not considered both the n-th coprime probability and its graph that ultimately became the target in this research. Hence, the newly defined terms are then used to find the generalizations of the n-th coprime probability and the n-th coprime graphs for some dihedral groups. The types and properties of the graphs are also discussed in this research.