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In a fuzzy graph G, a fuzzy coalition is formed by two disjoint vertex sets V1 and V2, neither of which is a strongly dominating set, but the union V1∪V2 forms a strongly dominating set. A fuzzy coalition partition of G is defined as Π={V1,V2,⋯,Vk}, where each set Vi either forms a singleton strongly dominating set or is not a strongly dominating set but forms a fuzzy coalition with another non-strongly dominating set in Π. A fuzzy graph with such a fuzzy coalition partition Π is called a fuzzy coalition graph, denoted as FG(G,Π). The vertex set of the fuzzy coalition graph consists of {V1,V2,⋯,Vk}, corresponding one-to-one with the sets of Π, and the two vertices are adjacent in FG(G,Π) if and only if Vi and Vj are fuzzy coalition partners in Π. This study demonstrates how fuzzy coalition graphs can model and optimize cybersecurity collaborations across critical infrastructures in smart cities, ensuring comprehensive protection against cyber threats. This study concludes that fuzzy coalition graphs offer a robust framework for optimizing collaboration and decision-making in interconnected systems like smart cities.

The single-valued neutrosophic set (SVNS) is a more common platform for representing the degree of truth, indeterminacy, and falsity memberships. Indeterminacy is an important aspect of SVNS, and the score function (SF) is important in ranking alternatives in decision-making scenarios. When the score and accuracy values for different SVNNs were equal, existing SFs were unable to rank the alternatives. To address the limitations, this paper introduces the perfect score function (PSF) as a solution for ranking single-valued neutrosophic numbers (SVNNs) without any errors and presents the concepts of strong and weak SVNSs. This article has emphasized the drawbacks of the existing SF in SVNNs. Additionally, we propose an optimized weighted geometric operator (OWGO) for SVNNs, and its properties such as idempotency, boundedness, monotonicity, and commutativity are explored. Furthermore, the decision-making approach has been depicted based on the proposed score function and OWGO to select the finest faculty for a university.

Let G^ be a simple connected graph with vertex set ϑG^ and edge set ξG^. The status of a vertex p∈ϑG^ is defined as ∑q≠pdp,q. A subset P of ϑG^ is called a status unequal dominating set (stu-dominating set) of G^; for every q∈ϑ−P, there exists p in P such that p and q are adjacent and stp≠stq. Among the most admired, extensively explored, and extensively discussed topics in the field of graph theory is graph coloring. This article recommends a new research study on the topic of incorporating stu-dominating set with coloring. Diverse coloring variants can be found in literature, some of which serve as inspiration is chromatic number, dominator coloring, color class domination, and colorful dominating sets. A novel study on the topic of incorporating stu-dominating set with coloring is suggested in this article. The stu-chromatic number, stu-dominator coloring, stu-color class domination, and colorful stu-domination have thus been explored and investigated. For each parameter introduced, the minimum cardinality of certain standard classes of graphs, bounds, and characterization results was explored.

Ocean iron fertilization is an approach to increase CO2 sequestration. The Indo-German iron fertilization experiment LOHAFEX was carried out in the Southern Ocean surrounding Antarctica in 2009 to monitor changes in bacterial community structure following iron fertilization-induced phytoplankton bloom of the seawater from different depths. 16S rRNA gene libraries were constructed using metagenomic DNA from seawater prior to and after iron fertilization and the clones were sequenced for identification of the major bacterial groups present and for phylogenetic analyses. A total of 4439 clones of 16S rRNA genes from ten 16S rRNA gene libraries were sequenced. More than 97.35% of the sequences represented four bacterial lineages i.e. Alphaproteobacteria, Gammaproteobacteria, Bacteroidetes and Firmicutes and confirmed their role in scavenging of phytoplankton blooms induced following iron fertilization. The present study demonstrates the response of Firmicutes due to Iron fertilization which was not observed in previous southern ocean Iron fertilization studies. In addition, this study identifies three unique phylogenetic clusters LOHAFEX Cluster 1 (affiliated to Bacteroidetes), 2 and 3 (affiliated to Firmicutes) which were not detected in any of the earlier studies on iron fertilization. The relative abundance of these clusters in response to iron fertilization was different. The increase in abundance of LOHAFEX Cluster 2 and Papillibacter sp. another dominant Firmicutes may imply a role in phytoplankton degradation. Disappearance of LOHAFEX Cluster 3 and other bacterial genera after iron fertilization may imply conditions not conducive for their survival. It is hypothesised that heterotrophic bacterial abundance in the Southern Ocean would depend on their ability to utilise algal exudates, decaying algal biomass and other nutrients thus resulting in a dynamic bacterial succession of distinct genera.

The notion of Fuzzy Graphs (FGs) and Intuitionistic Fuzzy Graphs (IFGs) is generalized in the Picture Fuzzy Graph (PFG), which is a more common platform for expressing the degree of positive, negative, and neutral membership functions. Picture Fuzzy Soft Graphs (PFSGs) are powerful mathematical tools for modeling real-world vagueness. The concept of vulnerability parameters of PFSG is provided in this research work by introducing the novel cardinality, domination number, integrity, and Domination Integrity (ÐÏ) of PFSG. Furthermore, a decision-making method for the PFSG has been presented using ÐÏ to suggest an algorithm for determining the ideal location for establishing a city diagnosis center. [ABSTRACT FROM AUTHOR]

In a simple, finite undirected graph G, a dominating set D is a subset of the vertex set V(G) whose closed neighbourhood is V(G). Many types of domination have been studied. The studies are based either on the nature of domination or the type of dominating set or the type of the complement of the dominating set. Interaction between dominating set and its complement is also considered. Fair domination is the domination where every vertex in the complement of a dominating set has equal number of neighbours in the dominating set. In this paper, a dominating set whose vertices have equal number of neighbours in the complement is the subject of study. The parameter γcof(G) is introduced and studied. [ABSTRACT FROM AUTHOR]