In this work, we associate a new topology to undirected graph G = (V, E) which may contain one isolated vertex or more and we named it Independent (non-adjacent vertices) Topology. A new sub-basis family to generate the Independent Topology is introduced on the set of n vertices V and for every vertex v of V the number of adjacent vertices is not greater than n − 2 (In simple graph we can say : for every vertex v of V, Δ(G) = n − 2, where Δ(G) is the maximum degree of vertices in a graph G). Then we give a fundamental step toward investigation of some properties of undirected graphs by their corresponding Independent Topology which we introduce in this work. Furthermore, an application to our new model (Independent Topology) are presented, that to carry out a framework in practical life like biomathematics (applied examples in biomathematics).

The aim of this article is to associate a tritopological space with undirected graph G = (V,E), i.e. three different topologies induced from the same graph or three different graphs. These tritopologies are the unique three proposed to associate topological spaces with graphs, the first which we proposed recently (Independent Topology) in 2020, the second is (Incidence Topology) proposed in 2018 and the third is proposed in 2013 (Graphic Topology). Then some properties of this tritopological space were investigated. Giving a fundamental step toward studying some properties of undirected graphs by their corresponding tritopological spaces is our motivation.