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This paper delves into the exploration of extremally disconnected spaces within the context of neutrosophic nano topological space. It introduces a novel space termed neutrosophic nano mixed space, created through the fusion of neutrosophic nano minimal structure and neutrosophic nano topology. The primary objective is to investigate extremal disconnectedness within this new space, shedding light on its properties and implications. Furthermore, the research examines the characterization of various types of open sets within the neutrosophic nano mixed space. As an extension, we gave a bio-mathematical application of neutrosophic nano extremal disconnectedness.

The aim of this paper was to generate nanotopological structure on the power set of vertices of simple digraphs using new definition neighbourhood of vertices on out linked of digraphs. Based on the neighbourhood we define the approximations of the subgraphs of a graph. A new nanotopological graph reduction to symbolic circuit analysis is developed in this paper. By means of structural equivalence on nanotopology induced by graph we have framed an algorithm for detecting patent infringement suit.

The main idea of this paper is to generate supra and infra topologies from simple undirected graphs. For this, we have introduced two new operators namely supra and infra operators which are defined on the power set of the vertex set of a graph. Moreover, we have also proved that the supra operator satisfying Kuratowski’s closure axiom will yield a topology. Further it was extended to develop the concept of connectedness and separation axioms on G-supra and G-infra spaces.

The main idea of this article is to define a fuzzy crisp set, intuitionistic crisp set and neutrosophic crisp set from simple digraphs. These sets have their own impact to generate the subbasis which in turn yields topological spaces. Moreover, an attempt has been made to extend our concept in induced subgraphs that lead us to relative topology. We have also formalized the structural equivalence of the isomorphic graphs and the topologies induced by them. A comparison between topologies has been made for some types of connected digraphs. Also, we have defined separation axioms on digraphs and related them to the topological separation axioms.