Topologies Emanating From Graphs

Title

Topologies Emanating From Graphs

Creator

Aniyan,Achu

Contributor

N K, Sudev

Description

A topology on a set is a collection of its subsets, including the set itself and the empty set, which is closed under union and fnite intersections. This dissertation introduces the notions such as graph topology, spanning graph topology, generalised graph topology, and generalised spanning graph topology by considering subgraphs and spanning subgraphs of a graph. Analogous to the set-theoretic notion, a graph topology is a collection of subgraphs of a given graph, including the null graph K0 and the graph itself, that is closed under newlineany union and any intersection. At the same time, a spanning graph topology of a graph is a collection of spanning subgraphs, including the spanning empty graph Nn, where n is the order of the graph and the graph G, which is closed under any union and any intersection. The topological concepts such as open sets, closed sets, base, subbase, neighbourhood, interior, subspace, and connectedness of spaces are extended to graph topology and spanning graph topology. In order to study the closed graphs in the above-mentioned graph topologies, two new graph complements are introduced in these graph topologies, such as decomposition and neighbourhood complements, to defne decomposition closed newlineand neighbourhood closed graphs. The decomposition complement is defned with respect to the edge set and the neighbourhood complement with respect to the vertex set. Since all the members of a spanning graph topology have the same vertex set, the neighbourhood closed graphs are described in terms of the edge set. The notion and characteristics of subspaces of both these graph topologies are defned, and the properties of closed graphs in these subspaces are also studied. Connectedness in topology holds a prominent role and applications in various felds of mathematics. The idea of connectedness is extended newlineto these graph topologies, and the same is characterised in the context of graph newlinetopology.

Source

Author's Submission

Date

2023-01-01

Publisher

Christ(Deemed to be University)

Subject

Mathematics and Statistics

Rights

Open Access

Relation

61000242

Format

PDF

Language

English

Type

PhD

Identifier

http://hdl.handle.net/10603/511502