Ratio Mathematica Volume 48, 2023
Topology via Graph Ideals
P.Gnanachandra*
K.Lalithambigai†
Abstract
The study of ideal topological space has started since 1933 and till
date it is being developed by several mathematicians. Various classes
of open sets, different types of operators and exploration of elemen-
tary topological results in ideal topological spaces have been dis-
cussed in various research papers. Methods of generating topolo-
gies using various relations have been explored by many researchers.
Many researchers explored the methods of inducing topologies via
graphs. This paper, introduces the notions of graph ideals, graph local
function and characterizes some of their properties. It also describes
a method of generating a new graph topology on the vertex set of a
graph from the graph adjacency topology using Kuratowski closure
operator and depicts the nature of open sets with respect to the new
topology. Further, it explores the condition for compatibility of the
graph adjacency topology with graph ideal.
Keywords: graph ideal, open neighbourhood function, graph local
function, graph adjacency topological space
2020 AMS subject classifications: 54H99, 57M15, 54A10, 54A05 1
*Centre for Research and Post Graduate Studies in Mathematics, Ayya Nadar Janaki Am-
mal College, Sivakasi (affiliated to Madurai Kamaraj University, Madurai), India; pgchan-
dra07@gmail.com.
†Sri Kaliswari College, Sivakasi (affiliated to Madurai Kamaraj University, Madurai), India;
lalithambigaimsc@gmail.com.
1Received on November 14, 2022. Accepted on July 9, 2023. Published on August 1, 2023. DOI:
10.23755/rm.v39i0.956. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is
published under the CC-BY licence agreement.
P.Gnanachandra, K.Lalithambigai
1 Introduction
Topology is concerned with the properties of a geometric object that are pre-
served under continuous deformations, such as stretching, twisting, crumpling
and bending. For a very long time, it was believed that abstract topological struc-
tures have limited applications in the generalization of real line and complex plane
or some connections to Algebra and other branches of Mathematics. Further, it
seems that there is a large gap between these structures and real life applications.
Generating topologies by relations and the representation of topological concepts
through binary relations will narrow the gap between Topology and its applica-
tions. A relation on graph represents a key for bridging Graph Theory and topo-
logical structures. The relation induces new types of topological structures to
the graph. In 1967, J.W.Evans et al.[5] showed that there is a one-to-one corre-
spondence between the labelled transitive directed graph with n points and the
labelled topologies on n points. In 1967, S.S.Anderson and G.Chartrand[1] inves-
tigated the lattice-graph of the topologies of transitive directed graphs presented
by J.W.Evans et al.[5]. In 2010, C.Marijuan[13] studied the relation between di-
rected graphs and finite topologies. In 2013, S.M.Amiri et al.[2] induced a topol-
ogy on vertex set of undirectd graph. In 2018, A.Kilicman and K.Abdulkalex[8]
associated an incidence topology with vertex set of simple graphs without iso-
lated vertices. In 2018, Shokry Nada et al.[15] generated topologies using the
post class relations on the vertex set of graphs and discussed some of its appli-
cations in the biomedical field. K.Lalithambigai and P.Gnanachandra in [12], de-
scribed a method of generating topology using adjacency, incidence relations on
vertex set of graphs and studied the properties of closure and interior of vertex set
of subgraphs in the graph adjacency topological space. Further, K.Lalithambigai
and P.Gnanachandra in [11], introduced graph grills and explored the properties
of topologies induced by graph grills on vertex set of graphs. The topic of ideals
in general topological spaces is treated in the classic text by Kuratowski[9],[10]
and also by R.Vaidyanathaswamy in [16],[17]. The properties which a topolog-
ical space possess locally and the conditions through which those properties be-
come a global one was studied by O.Njastad in 1966 [14]. Generation of new
topologies from the old one via ideals in general topological space was studied
by D.Jankovic and T.R.Hamlett[7]. Ideal on a topological space can be used to
study the similarity of two structures. Till now, scholars have mostly contributed
to the study of ideals in general topological space. It can be noticed that most
of the real life problems can be modelled as a graph and can be solved using the
graph theory concepts. The aim of this paper is to bridge the gap between abstract
and concrete concepts in ideal topological spaces, by defining graph ideals and
graph local functions in graph adjacency topological space. This paper explores
the basic facts of graph local function and describes the method of generating a
Topology via Graph Ideal
new topology from the older via graph ideals. Further, this paper characterizes the
nature of the open sets of the new topology in terms of the closure operator and
investigates the compatibility of graph adjacency topology with the graph ideal.
2 Preliminaries
Fundamental definitions and preliminaries of graph theory and topology may
be found in the sources [3],[4],[6].
A graph G consists of a pair (V(G),X(G)), where V(G) is a nonempty finite set
whose elements are called vertices and X(G) is a set of unordered pairs of dis-
tinct elements of V(G). The elements of X(G) are called edges of the graph G. An
edge joining a vertex to itself is called a loop. Edges joining the same vertices
are called multiple edges. A graph without loops and multiple edges is called a
simple graph. A graph G is called a bipartite graph if V can be partitioned into
two disjoint subsets V1 and V2 such that every edge of G joins a vertex of V1 to a
vertex of V2. (V1,V2) is called a bipartition of G. If further G contains every edge
joining the vertex of V1 to the vertex of V2, then G is called a complete bipartite
graph. If V1 contains m vertices and V2 contains n vertices then the complete bi-
partite graph G is denoted by Km,n. It should be noticed that K1,m is called a star
for m⩾1. The degree of a vertex v in a graph G is the number of edges incident
with v and it is denoted by deg(v). A graph in which degree of every vertex are the
same is called a regular graph. A vertex of degree 0 is called an isolated vertex.
The concept of ideals in topological spaces has been studied by Kuratowski[9]
and Vaidyanathaswamy[14].
An ideal on a topological space (X, τ) is a non empty collection I of subsets of X
satisfying the following two conditions: (i) If A ∈ I and B ⊂ A, then B ∈ I
(ii) If A ∈ I and B ∈ I, then A ∪ B ∈ I
An ideal topological space is a topological space (X, τ) with an ideal I on X, and
is denoted by (X, τ, I)
Throughout the paper, the graphs under discussion are the simple undirected
graphs which are not star graphs.
3 Graph Ideal and Graph Local Function
In this section, graph ideal and graph local function in a graph adjacency
topological space are defined with illustrations.
Definition 3.1. Let G = (V(G),X(G)) be a graph. For v ∈ V(G), the neighbourhood
set Nv of v is defined as Nv = {u ∈ V (G) : uv ∈ X(G)}.
P.Gnanachandra, K.Lalithambigai
Definition 3.2. Let G = (V(G),X(G)) be a graph without isolated vertices. Define
SN as the family of Nv for all v ∈ V(G), i.e., SN = {Nv : v ∈ V (G)}. Then SN
forms a subbase for a topology TA on V(G) and the pair (V(G),TA) is called graph
adjacency topological space. If W is a vertex induced subgraph of G, then the
closure of V(W) is defined by cl(V(W)) = V(W) ∪{v ∈ V (G) : Nv ∩ V (W) ̸= ϕ}
and the interior of V(W) is defined by int(V(W)) = {v ∈ V (G) : Nv ⊆ V (G)}.
Definition 3.3. Let G = (V(G),X(G)) be a graph for which P(V ) and P(X) are the
power sets of V(G) and X(G) respectively. The set I = {G′ : G′ = (V ′, X′ ), where
V
′ ⊆ V, X′ ⊆ X} is said to be a graph ideal on a graph adjacency topological
space (V(G),TA) if it satisfies the following two conditions
i) if G′ and G′′ ∈ I, then G′ ∪ G′′ ∈ I
ii) if G′ is a subgraph of G′′ and G′′ ∈ I, then G′ ∈ I.
Example 3.1. Consider the following graph
1
2
3
4
5
e
1
e 2
e 4
e
3
e
5
e
6
Figure. 3.1
For the graph in Figure 3.1,
I = {({1, 2}, {e1}), ({1}, ϕ), ({2}, ϕ), ({1, 2}, ϕ)} is a graph ideal.
I = {({1, 2, 3}, {e2}), ({4, 5}, {e6}), ({1, 2, 3, 4, 5}, {e2, e6}), ({1, 2}, {e1}) is not
a graph ideal, since, ({1}, ϕ) /∈ I, ({2}, ϕ) /∈ I, ({3}, ϕ) /∈ I, ({4}, ϕ) /∈
I, ({5}, ϕ) /∈ I.
Definition 3.4. Let (V (G), TA) be a graph adjacency topological space such that
v ∈ V(G). The open neighbourhood system at v denoted by N(v) is defined as N(v)
= {U ∈ TA : v ∈ U}.
Definition 3.5. Let (V (G), TA) be a graph adjacency topological space with a
graph ideal I. Let W be a subgraph of G. Then (V (W))∗(I, TA) = {v ∈ V (G) :
for every U ∈ N(v), V (W) ∩ U ̸= V (G′ ) for any G′ ∈ I} is called the graph
local function of V(W) with respect to I and TA.
Topology via Graph Ideal
Example 3.2. Consider the following graph
1
2
3
4
5
6
e 1
e
2
e
4
e3
Figure 3.2
Graph local function of a subgraph of the graph in Figure 3.2 is illustrated below:
SN = {{4, 5}, {6}, {5}, {1}, {1, 3}, {2}}.
B = {ϕ, {4, 5}, {6}, {5}, {1}, {1, 3}, {2}}.
TA = {ϕ, {4, 5}, {6}, {5}, {1}, {1, 3}, {2}, {4, 5, 6}, {4, 5, 1}, {1, 3, 4, 5}, {2, 4, 5}, {5, 6}, {1, 6},
{1, 3, 6}, {2, 6}, {1, 5}, {1, 3, 5}, {2, 5}, {1, 2}, {1, 5, 6}, {1, 4, 5}, {1, 4, 5, 6}, {1, 5, 6}, {1, 3, 5, 6},
{1, 3, 4, 5, 6}, {2, 4, 5, 6}, {2, 4, 5, 1}, {1, 2, 3, 4, 5}, {2, 5, 6}, {1, 2, 6}, {1, 2, 3, 6}, {1, 2, 5},
{1, 2, 3, 5}, {1, 2, 5, 6}, {1, 2, 4, 5}, {1, 2, 4, 5, 6}, {1, 2, 5, 6}, {1, 2, 3, 5, 6}, {1, 2, 3, 4, 5, 6}}.
Let I = {({1}, ϕ), ({5}, ϕ), ({1, 5}, ϕ), ({1, 5}, e2)} be a graph ideal.
Let W = ({1, 2, 3}, ϕ) be a subgraph of the given graph. Then (V (W))∗(I, TA) =
{2, 3}.
Note 3.6. 1. Let G be a graph with n vertices. For each i = 1,2, ...n, {({vi}, ϕ)}
is a graph ideal. In this case, for any subgraph W , (V (W))∗(I, TA) =
cl(V (W)).
2. Let P(G) be the collection of all subgraphs of a graph G. Clearly P(G) is
a graph ideal. In this case, for any subgraph W, (V (W))∗(I, TA) = ϕ.
4 Facts Concerning Graph Local Function
This section describes some basic facts concerning the graph location function
which are useful in the generation of new topology from the old one.
Theorem 4.1. Let (V (G), TA) be a graph adjacency topological space with graph
ideals I and J . Let H and W be subgraphs of G. Then
1. V(H) ⊆ V(W) ⇒ (V(H))∗ ⊆ (V (W))∗
2. I ⊆ J ⇒ (V (W))∗(J ) ⊆ (V (W))∗(I)
P.Gnanachandra, K.Lalithambigai
3. (V(W))∗ ⊆ cl(V(W))
4. ((V(W))∗)∗ ⊆ (V (W))∗
5. (V(H ∪ W))∗ = (V (H))∗ ∪ (V (W))∗
6. (V (H))∗ − (V (W))∗ ⊆ (V (H) − V (W))∗
7. U ∈ TA ⇒ U ∩ (V (W))∗ = U ∩ (U ∩ V (W))∗ ⊆ (U ∩ V (W))∗
8. (V(W))∗(I) ∪ (V (W))∗(J ) = (V (W))∗(I ∩ J ).
9. (V (H) ∩ V (W))∗ ⊆ (V (H))∗ ∩ (V (W))∗.
10. I = V(G
′
) for some G
′ ∈ I ⇒ (V (H) ∪ I)∗ = (V (H))∗ = (V (H) − I)∗.
11. (V(H))∗ - ((V(H))∗)∗ ⊆ (V (H) − (V (H))∗)∗.
Proof. 1. Let v /∈ (V (W))∗.
Hence there exists U ∈ N(v) such that V (W) ∩ U = V (G′), for some G′ ∈ I.
Since V (H) ⊆ V (W), V (H) ∩ U ⊆ V (W) ∩ U.
Hence there exists U ∈ N(v) such that V (H) ∩ U = V (G′), for some G′ ∈ I.
So v /∈ (V (H))∗. Hence (V (H))∗ ⊆ (V (W))∗.
2. Let v ∈ (V (W))∗(J ).
Hence, for every U ∈ N(v), V (W) ∩ U ̸= V (G′), for any G′ ∈ J .
Since I ⊆ J , V (W) ∩ U ̸= V (G′), for any G′ ∈ I.
Hence for every U ∈ N(v), V (W) ∩ U ̸= V (G′), for any G′ ∈ I.
Hence (V (W))∗(J ) ⊆ (V (W))∗(I)
3. Let v ∈ (V (W))∗.
Hence for every U ∈ N(v), V (W) ∩ U ̸= V (G′), for any G′ ∈ I.
Thus V (W) ∩ Nv ̸= ϕ and v ∈ cl(V (W)).
Hence (V (W))∗ ⊆ cl(V (W)).
4. Let v ∈ ((V (W))∗)∗.
Hence for every U ∈ N(v), (V (W))∗ ∩ U ̸= V (G′), for any G′ ∈ I.
For every U ∈ N(v), V (W) ∩ U ̸= V (G′), for any G′ ∈ I.
So v ∈ (V (W))∗ and ((V (W))∗)∗ ⊆ (V (W))∗.
5. Let v /∈ (V (H))∗ ∪ (V (W))∗.
Then v /∈ (V (H))∗ and v /∈ (V (W))∗.
So there exists U1 ∈ N(v) such that V (H) ∩ U1 = V (G
′
), for some G′ ∈ I and
U2 ∈ N(v) such that V (W) ∩ U2 = V (G
′′
), for some G′′ ∈ I.
Hence (V (H) ∪ V (W)) ∩ U = V (G′′′), where U = U1 ∪ U2 and V (G
′′′
) =
V (G′) ∪ V (G′′).
So there exists U ∈ N(v) such that (V (H) ∪ V (W)) ∩ U = V (G′′′), for some
G′′′ ∈ I.
Topology via Graph Ideal
Hence v /∈ (V (H ∪ W))∗ and (V (H ∪ W))∗ ⊆ (V (H))∗ ∪ (V (W))∗ .
Thus V (H) ⊆ V (H ∪ W) and V (W) ⊆ V (H ∪ W).
By (1), (V (H))∗ ⊆ (V (H ∪ W))∗ and (V (W))∗ ⊆ (V (H ∪ W))∗.
Therefore (V (H))∗ ∪ (V (W))∗ ⊆ (V (H ∪ W))∗.
Hence (V (H))∗ ∪ (V (W))∗ = (V (H ∪ W))∗.
6. Let v /∈ (V (H) − V (W))∗.
Hence there exists U ∈ N(v) such that (V (H) − V (W)) ∩ U = V (G′), for some
G′ ∈ I.
So (V (H) ∩ U) − (V (W) ∩ U) = V (G′), for some G′ ∈ I.
Thus (V (H) ∩ U) = V (G′) and V (W) ∩ U ̸= V (G′), for some G′ ∈ I.
So v /∈ (V (H))∗ and v ∈ (V (W))∗.
Thus v /∈ (V (H))∗ − (V (W))∗.
Hence (V (H))∗ − (V (W))∗ ⊆ (V (H) − V (W))∗.
7. Let U ∈ TA and v ∈ U ∩ (V (W))∗.
v ∈ U and v ∈ (V (W))∗.
v ∈ U and U ∈ TA ⇒ U ∈ N(v).
v ∈ (V (W))∗ ⇒ for every U1 ∈ N(v), V (W) ∩ U1 ̸= V (G
′
), for any G′ ∈ I.
U ∩ (V (W) ∩ U1) ̸= V (G
′
), for any G′ ∈ I.
(U ∩ V (W)) ∩ U1 ̸= V (G
′
), for any G′ ∈ I.
Hence v ∈ (U ∩ V (W))∗.
So v ∈ U ∩ (U ∩ V (W))∗.
U ∩ (V (W))∗ ⊆ U ∩ (U ∩ V (W))∗.
By reserving the above steps, it follows that U ∩ (U ∩ V (W))∗ ⊆ U ∩ (V (W))∗.
So U ∩ (V (W))∗ = U ∩ (U ∩ V (W))∗.
Moreover, U ∩ (U ∩ V (W))∗ ⊆ (U ∩ V (W))∗.
8. Since I ∩ J ⊆ I and I ∩ J ⊆ J , (V (W))∗(I) ⊆ (V (W))∗(I ∩ J ) and
(V (W))∗(J ) ⊆ (V (W))∗(I ∩ J ).
Hence (V (W))∗(I) ∪ (V (W))∗(J ) ⊆ (V (W))∗(I ∩ J ).
Let v ∈ (V (W))∗(I ∩ J ). So, for every U ∈ N(v), V (W) ∩ U ̸= V (G′), for any
G′ ∈ I ∩ J .
For every U ∈ N(v), V (W)∩U ̸= V (G′), for any G′ ∈ I or V (W)∩U ̸= V (G′),
for any G′ ∈ J .
So v ∈ (V (W))∗(I) or v ∈ (V (W))∗(J ) and so v ∈ (V (W))∗(I)∪(V (W))∗(J ).
(V (W))∗(I ∩ J ) ⊆ (V (W))∗(I) ∪ (V (W))∗(J ).
So (V (W))∗(I ∩ J ) = (V (W))∗(I) ∪ (V (W))∗(J ).
9. Since V (H) ∩ V (W) ⊆ V (H) and V (H) ∩ V (W) ⊆ V (W), by (1),
(V (H) ∩ V (W))∗ ⊆ (V (H))∗ and (V (H) ∩ V (W))∗ ⊆ (V (W))∗.
Hence (V (H) ∩ V (W))∗ ⊆ (V (H))∗ ∩ (V (W))∗.
10. Since V (H) ⊆ V (H) ∪ I and V (H) − I ⊆ V (H), by (1),
(V (H))∗ ⊆ (V (H) ∪ I)∗ and (V (H) − I)∗ ⊆ (V (H))∗.
Now, v /∈ (V (H) − I)∗
P.Gnanachandra, K.Lalithambigai
⇒ there exists U ∈ N(v) such that U ∩ (V (H) − I) = V (G′), for some G′ ∈ I.
⇒ there exists U ∈ N(v) such that U ∩ (V (H) − V (G′′)) = V (G′), for some
G′, G′′ ∈ I.
So there exists U ∈ N(v) such that (U ∩ V (H)) − (U ∩ V (G′′)) = V (G′), for
some G′, G′′ ∈ I.
This implies that there exist U ∈ N(v) such that U ∩ V (H) = V (G′), for some
G′ ∈ I.
⇒ v /∈ (V (H))∗.
So (V (H))∗ ⊆ (V (H) − I)∗.
Let v ∈ (V (H) ∪ I)∗.
For every U ∈ N(v), U ∩ (V (H) ∪ I) ̸= V (G′), for any G′ ∈ I.
For every U ∈ N(v), (U ∩ V (H)) ∪ (U ∩ I) ̸= V (G′), for any G′ ∈ I.
But I = V (G′′), for some G′′ ∈ I.
So U ∩ V (H) ̸= V (G′′′), for any G′′′ ∈ I.
So v ∈ (V (H))∗.
Hence (V (H) ∪ I)∗ ⊆ (V (H))∗.
So (V (H) ∪ I)∗ = (V (H))∗ = (V (H) − I)∗.
11. v ∈ (V (H))∗ − ((V (H))∗)∗.
So v ∈ (V (H))∗ and v /∈ ((V (H))∗)∗.
For every U ∈ N(v), U ∩ V (H) ̸= V (G′), for any G′ ∈ I and there exists
U1 ∈ N(v) such that U1 ∩ (V (H))∗ = V (G
′′
), for some G′′ ∈ I .
For every U ∈ N(v), (U ∩ V (H)) − (U ∩ (V (H))∗) ̸= V (G′′′), for any G′′′ ∈ I.
For every U ∈ N(v), U ∩ (V (H) − (V (H))∗) ̸= V (G′′′), for any G′′′ ∈ I.
v ∈ (V (H) − (V (H))∗)∗.
So (V (H)∗ − ((V (H))∗)∗ ⊆ (V (H) − (V (H))∗)∗.
5 Topology via Graph Ideals
This section describes a method of generating a new topology from the old
one using Kuratowski closure operator cl∗.
Definition 5.1. Let (V (G), TA) be a graph adjacency topological space with a
graph ideal I. Let W be a subgraph of G. Define cl∗(V (W)) = V (W)∪(V (W))∗
and T ∗A(I) = {V (H) ⊆ V (G) : cl
∗(V (G) − V (H)) = V (G) − V (H)}. T ∗A(I) is
called the graph adjacency topology generated by cl∗. When there is no ambiguity,
it is denoted as T ∗A .
Example 5.1. Consider the following graph
Topology via Graph Ideal
1
2
34
e
1
e
2
e 4
e3
SN = {{2}, {1, 3, 4}, {2, 4}, {2, 3}}
B = {ϕ, {2}, {1, 3, 4}, {2, 4}, {2, 3}, {4}, {3}}
TA = {ϕ, {2}, {1, 3, 4}, {2, 4}, {2, 3}, {4}, {3}, {3, 4}, {2, 3, 4}, {1, 2, 3, 4}}
I = {({1, 2}, {e1}), ({1}, ϕ), ({2}, ϕ), ({1, 2}, ϕ)}
N(1) = {{1, 3, 4}, {1, 2, 3, 4}}, N(2) = {{2}, {2, 4}, {2, 3}, {2, 3, 4}, {1, 2, 3, 4}},
N(3) = {{1, 3, 4}, {2, 3}, {3}, {3, 4}, {2, 3, 4}, {1, 2, 3, 4}},
N(4) = {{1, 3, 4}, {2, 4}, {4}, {3, 4}, {2, 3, 4}, {1, 2, 3, 4}}
T ∗A = {ϕ, {3}, {4}, {3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}.
Observation 5.2. 1. If I = {({vi}, ϕ)} is a graph ideal, then (V (W))∗(I, TA) =
cl(V (W)) for every V (W) ⊆ V (G).
Hence cl∗(V (W)) = cl(V (W)).
T ∗A = {V (H) ⊆ V (G) : cl
∗(V (G) − V (H)) = V (G) − V (H)}
= {V (H) ⊆ V (G) : cl(V (G) − V (H)) = V (G) − V (H)}
= TA.
2. If P(G) is the collection of all subgraphs of a graph G. Then P(G) is
a graph ideal. Hence for any subgraph W, (V (W))∗(I, TA) = ϕ and
cl∗(V (W)) = V (W).
T ∗A = {V (H) ⊆ V (G) : cl
∗(V (G) − V (H)) = V (G) − V (H)}
= {V (H) ⊆ V (G) : V (G) − V (H) = V (G) − V (H)}
= P(G)
Hence T ∗A is a discrete topology.
3. For every graph ideal I, {({vi}, ϕ)} ⊆ I ⊆ P(G). Hence TA ⊆ T ∗A ⊆
discrete topology.
Proposition 5.1. If I and J are graph ideals such that I ⊆ J , then T ∗A(I) ⊆
T ∗A(J ).
Proof. I ⊆ J ⇒ (V (H))∗(J ) ⊆ (V (H))∗(I). Thus {V (H) ∪ (V (H))∗(J )} ⊆
{V (H) ∪ (V (H))∗(I)}. Therefore cl∗(V (H))(J ) ⊆ cl∗(V (H))(I). This means
cl∗(V (G) − V (H))(I) ⊆ cl∗(V (G) − V (H))(J ). Hence T ∗A(I) ⊆ T
∗
A(J ).
P.Gnanachandra, K.Lalithambigai
Definition 5.3. The derived set of V(H) in V(G), TA) denoted by (V (H))d is de-
fined by v ∈ (V (H))d if and only if (U − {v}) ∩ V (H) ̸= ϕ for every U ∈ N(v).
The derived set of V(H) in V(G), T ∗A) is denoted by (V (H))
d∗.
Example 5.2. Consider the following graph
1
5
2
4
3
e
1
e 2 e
4
e
5
e3
SN = {{1}, {5, 4}, {4}, {3, 2, 5}, {1, 2, 4}}
B = {ϕ, {1}, {5, 4}, {4}, {3, 2, 5}, {1, 2, 4}, {2}}
TA = {ϕ, {1}, {5, 4}, {4}, {3, 2, 5}, {1, 2, 4}, {2}, {1, 2, 4, 5}, {2, 5}, {2, 3, 4, 5}, {2, 4, 5}, {2, 4},
{1, 2, 3, 4, 5}}
Let V (H) = {3, 5}. Then (V (H))d = {3}.
Note 5.4. 1. v ∈ (V (H))d∗ ⇔ v ∈ cl∗(V (H) − {v})
⇔ v ∈ (V (H) − {v}) ∪ (V (H) − {v})∗
⇔ v ∈ (V (H) − {v})∗
⇔ for every U ∈ N(v), (V (H) − {v}) ∩ U ̸= V (G′), for any G′ ∈ I
Hence v ∈ (V (H))d∗ ⇔ for every U ∈ N(v), (V (H)−{v})∩U ̸= V (G′),
for any G
′ ∈ I.
2. (V (H))d∗ ⊆ (V (H))∗
Definition 5.5. A graph which has either an infinite number of vertices or edges
or both is called an infinite graph.
Definition 5.6. Let G be an infinite graph. Let If be the graph ideal of finite sub-
graphs of G and H be a subgraph of G. A vertex v is said to be an ω-accumulation
point of V(H) if and only if U ∩ V (H) is infinite for every U ∈ N(v). The set of
all ω-accumulation points of V(H) is denoted by (V (H))ω.
Observation 5.7. (V (H))ω = (V (H))∗(If).
Definition 5.8. Let G be an infinite graph. Let Ic be the graph ideal of countable
subgraphs of G and H be a subgraph of G. A vertex v is said to be a condensation
point of V(H) if U ∩ V(H) is uncountable for every U ∈ N(v). The set of all
condensation points of V(H) is denoted by (V (H))∗(Ic).
Topology via Graph Ideal
Definition 5.9. A graph adjacency topological space (V (G), TA) is said to be a T1
space if for each pair of distinct vertices vi and vj, vi belong to every U ∈ N(vi)
for which vi does not belong to any U ∈ N(vj) and vj belongs to every U ∈ N(vj)
and vj does not belong to any U ∈ N(vi).
Remark 5.1. 1. Let If be a graph ideal of finite subgraphs of G. Since ({vi}, ϕ) ∈
If for each vi ∈ V (G), (V (H))∗ = (V (H))d∗.
Also (V (H))∗ = (V (H))ω. Hence ω-accumulation points of V(H) in (V (G), TA)
are precisely the limit points of V(H) in (V (G), T ∗A(If)). Since ({vi}, ϕ) ∈
If, ({vi}, ϕ)∗ = ϕ. So cl∗({vi}) = {vi}.Hence (V (G), T ∗A) is T1.
In T1 spaces, ω-accumulation point and limit point coincides. Hence the set
of ω-accumulation points of V(H) in (V (G), TA) and (V (G), T ∗A) coincide.
(V (H))∗(If) = (V (H))d iff TA = T ∗A(If)) iff (V (G), TA) is T1.
2. Let Ic be a graph ideal of countable subgraphs of G. Since (V (H))∗ is the
set of condensation points of V(H) and (V (H))∗ = (V (H))d∗, the conden-
sation points of V(H) in (V(G), TA) are precisely the limit points of V(H) in
(V (G), T ∗A(Ic)).
Definition 5.10. Let H be a subgraph of a graph G. We say that V(H) is closed
and discrete in (V (G), TA) if and only if (V (H))d = ϕ.
Lemma 5.1. Let (V (G), TA) be a graph adjacency topological space with a graph
ideal I. If I ∈ I, then I is closed and discrete in (V (G), T ∗A).
Proof. I ∈ I ⇒ I∗ = ϕ ⇒ Id∗ = ϕ.
Remark 5.2. Let (V (G), TA) be a graph adjacency topological space. Let Icd =
{H : H is a subgraph of G and (V (H))d = ϕ}.
Let us prove Icd is a graph ideal.
H1, H2 ∈ Icd ⇒ (V (H1))d = ϕ and (V (H2))d = ϕ.
Hence for every v ∈ V (G), (U −{v})∩V (H1) = ϕ and (U −{v})∩V (H2) = ϕ.
So for every v ∈ V (G), (U − {v}) ∩ V (H1 ∪ H2) = ϕ.
So H1 ∪ H2 ∈ Icd.
Let H1 ∈ Icd and H2 be a subgraph of H1.
H1 ∈ Icd ⇒ (V (H1))d = ϕ.
Hence for every v ∈ V (G), (U − {v}) ∩ V (H1) = ϕ and so for every v ∈
V (G), (U − {v}) ∩ V (H2) = ϕ.
So (V (H2))d = ϕ which implies H2 ∈ Icd .
Icd is a graph ideal. Also (V (H))d ⊆ (V (H))∗. Hence T ∗A = TA.
Lemma ?? implies that Icd is the largest graph ideal with the property that T ∗A =
TA.
Finally, (V (H))∗ = (V (H))d iff (V (G), TA) is T1.
P.Gnanachandra, K.Lalithambigai
Example 5.3. Let (V (G), TA) be a graph adjacency topological space. Let I =
{G′ : G′ is a subgraph of G and int(cl(V (G′))) = ϕ}, i.e., I is the collection of
nowhere dense subgraphs of (V (G), TA).
Let G
′
, G
′′ ∈ I. Hence int(cl(V (G′))) = ϕ and int(cl(V (G′′))) = ϕ.
Now, int(cl(V (G
′ ∪ G′′))) = int(cl(V (G′) ∪ cl(V (G′′))) ⊇ int(cl(V (G′)) ∪
int(cl(V (G
′′
)) = ϕ.
Hence int(cl(V (G
′ ∪ G′′))) ⊇ ϕ.
So I is not a graph ideal.
6 The Open sets of T ∗A
Let (V (G), TA) be a graph adjacency topological space with a graph ideal
I. Let H be a subgraph of G. Then V(H) is said to be T ∗A -closed if and only if
((V (H))∗ ⊆ V (H). Hence V(H) is T ∗A -open if and only if V(G) - V(H) is T
∗
A -
closed, i.e., if and only if (V (G) − V (H))∗ ⊆ V (G) − V (H), i.e., if and only if
V (H) ⊆ V (G) − (V (G) − V (H))∗.
Hence V(H) is T ∗A -open if and only if v ∈ V (H) ⇒ v /∈ (V (G) − V (H))
∗. So
V(H) is T ∗A -open if and only if v ∈ V (H) ⇒ there exists U ∈ N(v) such that
U ∩ (V (G) − V (H)) = V (G′), for some G′ ∈ I.
Let W = U ∩ (V (G) − V (H)). V(H) is T ∗A -open if and only if, for v ∈ V (H)
there exists U ∈ N(v) such that W = V (G′), for some G′ ∈ I.
If v ∈ V (H) then v /∈ V (G)−V (H). So v /∈ W . Also v ∈ U. Hence v ∈ U −W .
V(H) is T ∗A -open if and only if, for v ∈ V (H), there exists U ∈ N(v) such that
v ∈ U − W and W = V (G′) for some G′ ∈ I.
Let β(I, T ∗A) = {U − W : U ∈ TA, W = V (G
′
) for some G
′ ∈ I}. β(I, T ∗A) is a
basis for T ∗A .
Note 6.1. If I and J are graph ideals then I ∨ J = {I ∪ J : I ∈ I, J ∈ J } is
also a graph ideal.
Theorem 6.1. Let (V (G), TA) be a graph adjacency topological space with a
graph ideals I and J . Let H be a subgraph of G. Then (V (H))∗(I ∨ J , TA) =
(V (H))∗(I, T ∗A(J )) ∩ (V (H))
∗(J , T ∗A(I)).
Proof. Let v /∈ (V (H))∗(I ∨ J , TA).
Hence there exists U ∈ N(v) such that U ∩V (H) = V (G′), for some G′ ∈ I ∨J .
Let G
′′ ∈ I and G′′′ ∈ J such that U ∩ V (H) = V (G′′′′) where G′′′′ = G′′ ∪ G′′′ .
Assume that V (G
′′
) ∩ V (G′′′) = ϕ.
Hence (U ∩ V (H)) − V (G′′) = V (G′′′) and (U ∩ V (H)) − V (G′′′) = V (G′′).
So (U −V (G′′))∩V (H) = V (G′′′), where G′′′ ∈ J and (U −V (G′′′))∩V (H) =
V (G
′′
), where G
′′ ∈ I.
Topology via Graph Ideal
⇒ v /∈ (V (H))∗(J , T ∗A(I)) or v /∈ (V (H))
∗(I, T ∗A(J )).
⇒ v /∈ (V (H))∗(J , T ∗A(I)) ∩ (V (H))
∗(I, T ∗A(J )).
Hence (V (H))∗(I, T ∗A(J )) ∩ (V (H))
∗(J , T ∗A(I)) ⊆ (V (H))
∗(I ∨ J , TA).
v /∈ (V (H))∗(I, T ∗A(J )) implies that there exists U ∈ N(v) such that (U −
V (G
′′
)) ∩ V (H) = V (G′) for some G′ ∈ I, G′′ ∈ J .
Assume that V (G
′′
) ⊆ V (H). Hence U ∩ V (H) = V (G′) ∪ V (G′′) = V (G′′′),
where G
′′′ ∈ I ∨ J . So v /∈ (V (H))∗(I ∨ J , TA).
(V (H))∗(I ∨ J , TA) ⊆ (V (H))∗(I, T ∗A(J )).
Similarly, (V (H))∗(I ∨ J , TA) ⊆ (V (H))∗(J , T ∗A(I)).
(V (H))∗(I ∨ J , TA) ⊆ (V (H))∗(I, T ∗A(J )) ∩ (V (H))
∗(J , T ∗A(I)).
Hence (V (H))∗(I ∨ J , TA) = (V (H))∗(I, T ∗A(J )) ∩ (V (H))
∗(J , T ∗A(I)).
Remark 6.1. Given a graph adjacency topological space (V (G), TA) and a graph
ideal I, T ∗∗A = (T
∗
A(I))
∗(I) is a topology on V (G) and T ∗∗A is finer than T
∗
A .
Corollary 6.1. Let (V (G), TA) be a graph adjacency topological space with a
graph ideal I. Then (V (H))∗(I, TA) = (V (H))∗(I, T ∗A) and hence T
∗
A = T
∗∗
A .
Proof. (V (H))∗(I, TA) =
(V (H))∗(I ∨ I, TA) = (V (H))∗(I, T ∗A(I)) ∩ (V (H))
∗(I, T ∗A(I))
= (V (H))∗(I, T ∗A(I))
T ∗∗A = T
∗
A(I))
∗(I) = T ∗A .
7 Compatibility of TA with I
This section introduces the definition of compatibility of TA with the graph
ideal I and explores its significance.
Definition 7.1. Let (V (G), TA) be a graph adjacency topological space with a
graph ideal I. The graph adjacency topology TA is said to be compatible with the
graph ideal I, denoted TA ∽ I, if the following holds for every vertex induced
subgraph H of G :
if for every v ∈ V (H), there exists a U ∈ N(v) such that U ∩ V (H) = V (G′),
for some G
′ ∈ I, then V (H) = V (G′′), for some G′′ ∈ I.
Example 7.1. Consider the following graph
P.Gnanachandra, K.Lalithambigai
1
4
3
2
e
1
e 2
e 4
SN = {{4}, {3}, {2, 4}, {1}}.
B = {ϕ, {4}, {3}, {2, 4}, {1}}.
TA = {ϕ, {4}, {3}, {2, 4}, {1}, {3, 4}, {1, 4}, {2, 3, 4}, {1, 3}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}}.
N(1) = {{1}, {1, 4}, {1, 3}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}}.
N(2) = {{2, 4}, {2, 3, 4}, {1, 2, 4}, {1, 2, 3, 4}}.
N(3) = {{3}, {3, 4}, {2, 3, 4}, {1, 3}, {1, 3, 4}, {1, 2, 3, 4}}.
N(4) = {{4}, {2, 4}, {3, 4}, {1, 4}, {2, 3, 4}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}}.
TA is compatible with the graph ideal I = {({1}, ϕ), ({2}, ϕ), ({1, 2}, ϕ)} and
not compatible with the graph ideal I = {({1}, ϕ), ({4}, ϕ), ({1, 4}, ϕ)}.
Observation 7.2. Let (V (G), TA) be a graph adjacency topological space with a
graph ideal I. Then the following are implied by TA ∽ I :
(a) For every vertex induced subgraph H of G, V (H) ∩ (V (H))∗ = ϕ ⇒
(V (H))∗ = ϕ.
(b) For every vertex induced subgraph H of G, (V (H) − (V (H))∗)∗ = ϕ.
(c) For every vertex induced subgraph H of G, (V (H)∩(V (H))∗)∗ = (V (H))∗.
(d) β(I, T ∗A) = {U − W : U ∈ TA, W = V (G
′
), for some G
′ ∈ I} = T ∗A .
Theorem 7.1. Let (V (G), TA) be a graph adjacency topological space with a
graph ideal I. Then the following are equivalent:
(1) TA ∽ I
(2) For every vertex induced subgraphs H of G, V (H)∩(V (H))∗ = ϕ ⇒ V (H) =
V (G
′
), for some G
′ ∈ I.
(3) For every vertex induced subgraphs H of G, V (H)−(V (H))∗ = ϕ or V (H)−
(V (H))∗ = V (G
′
), for some G
′ ∈ I.
(4) For every T ∗A closed subgraph H, V (H)−(V (H))
∗ = V (G
′
) for some G
′ ∈ I.
(5) For every vertex induced subgraphs H of G, if V(H) contains no induced sub-
graph W with V (W) ⊆ (V (W))∗, then V (H) = V (G′) for some G′ ∈ I.
Topology via Graph Ideal
Proof. (1) ⇒ (2) is obvious.
(2) ⇒ (3): For every vertex induced subgraphs H of G, V (H) − (V (H))∗ = ϕ or
(V (H)−(V (H))∗)∩(V (H)−(V (H))∗)∗ = ϕ. So by (2), V (H)−(V (H))∗ = ϕ
or V (H) − (V (H))∗ = V (G′), for some G′ ∈ I.
(3) ⇒ (4) is straightforward.
(4) ⇒ (1): Let H be a vertex induced subgraph of G and assume that for every v
∈ V(H), there exists U ∈ N(v) such that U ∩ V (H) = V (G′) for some G′ ∈ I.
Then V (H) ∩ (V (H))∗ = ϕ.
Since (V (H)∪(V (H))∗)∗ = (V (H))∗∪((V (H))∗)∗ ⊆ (V (H))∗∪V (H), V (H)∪
(V (H))∗ is T ∗A closed.
So by (4), (V (H) ∪ (V (H))∗) − (V (H) ∪ (V (H))∗)∗ = V (G′) for some G′ ∈ I.
But (V (H)∪(V (H))∗)−(V (H)∪(V (H))∗)∗ = (V (H)∪(V (H))∗)−((V (H))∗∪
(V (H))∗∗) = (V (H) ∪ (V (H))∗) − (V (H))∗ = V (H).
Hence V (H) = V (G
′
), for some G
′ ∈ I.
So TA ∽ I
(3) ⇒ (5): Let H be a vertex induced subgraph of G and assume that V(H) contains
no induced subgraph W with V (W) ⊆ (V (W))∗.
Since V (H)−(V (H))∗ = V (G′), for some G′ ∈ I, V (H)∩(V (H))∗ ⊆ (V (H)∩
(V (H))∗)∗, V (H) ∩ (V (H))∗ = ϕ.
So V (H) = V (H) − (V (H))∗ and V (H) = V (G′), for some G′ ∈ I.
(5) ⇒ (3) : Since (V (H)−(V (H))∗)∩(V (H)−(V (H))∗)∗ = ϕ, V (H)−(V (H))∗
contains no vertex induced subgraph W such that V (W) ⊆ (V (W))∗.
So by (5), V (H) − (V (H))∗ = V (G′) for some G′ ∈ I.
Theorem 7.2. Let (V (G), TA) be a graph adjacency topological space with a
graph ideal I and TA ∽ I. Let H be a vertex induced subgraph of G. If V(H) is
T ∗A closed then it is the union of V(W), where (V (W))
∗ ⊆ cl(V (W)), and V (G′)
for some G
′ ∈ I.
Proof. Let V(H) be T ∗A closed. Then (V (H))
∗ ⊆ V (H). So V (H) = (V (H) −
(V (H))∗) ∪ (V (H))∗.
By theorem ??(3), V (H) − (V (H))∗ = V (G′) for some G′ ∈ I.
Also (V (H))∗ ⊆ cl(V (H)).
So V(H)is the union of V(W), where (V (W))∗ ⊆ cl(V (W)), and V (G′) for some
G
′ ∈ I.
8 Conclusions
Graph ideal and graph local function in a graph adjacency topological space
have been defined and the basic facts concerning graph local function have been
P.Gnanachandra, K.Lalithambigai
proved. The method of generating a new topology from the older one using Ku-
ratowski closure operator via graph ideal has been discussed. Further, the char-
acteristics of the open sets of the new topology in terms of the closure operator
have been analyzed. New topologies generated by two different ideals have been
compared. The feature of graph location function based on the union of two ideals
has been studied. The compatibility of the graph adjacency topology with graph
ideal has been defined and the equivalent conditions for compatibility of the graph
adjacency topology with graph ideal have been investigated. Further, the results
in this paper are useful in the study of some new sets and topologies in graph
adjacency topological space with graph ideal. These concepts can be studied fur-
ther by stating graph local function of induced subgraphs of a graph with respect
to graph adjacency topology using graph prime ideal and graph principal ideal.
The open sets of topologies generated via prime ideals and principal ideals can be
compared and investigated in future studies.
Acknowledgement
We are overwhelmed to acknowledge our thankfulness to those who helped us
to put these ideas, well above the level of simplicity and into something concrete.
We would like to express our special thanks to Editor in Chief, Ratio Mathematica
for his time and efforts for the successful completion of this paper. His useful
advice and suggestions were really helpful to us. The generosity and expertise of
the reviewers have improved this study in innumerable ways and saved us from
many errors;those that inevitably remain are entirely our own responsibility. We
gratefully acknowledge the reviewers who stood by us and encouraged us to work
on this paper.
References
S.S.Anderson,G.Chartrand,The Lattice-graph of the Topology of a Transitive Di-
rected Graph,Mathematica Scandinavica,21(1967),105-109.
S.M. Amiri,A.Jafarzadeh,H.Khatibzadeh An Alexandroff Topology on
Graphs,Bulletin of Iranian Mathematical Society,39(2013),647-662
J.A.Bondy,U.S.R.Murthy,Graph Theory,Graduate Texts in Mathemat-
ics,Springer,Berlin,(2008).
Chartrand G.Lesniak L.Zhang P.,Textbooks in Mathematics(Graphs and Di-
graphs),Sixth Edition. Taylor and Francis Group.LLC.(2016).
Topology via Graph Ideal
J.W.Evans,F.Harary and M.S.Lynn,On the Computer Enumeration of Finite
Topologies, Commu. Assoc. Comp. Mach,(10)(1967),295-298.
James R.Munkres,Topology,Second Edition.Pearson Education,Inc.(2006)
D.Jankovic and T.R.Hamlett,New Topologies from old via Ideals,Amer. Math
Monthly,97(1990),295-310.
A.Kilicman,K.Abdulkalek,Topological Spaces Associated with simple
Graphs,Journal of Mathematical Analysis,9(4)(2018),44-52.
K.Kuratowski,Topology,I.Warszawa,(1933).
K.Kuratowski,Topology,Vol.I. Academic Press,New York,(1966).
K.Lalithambigai and P.Gnanachandra, Topologies Induced by Graph Grills on
Vertex Set of Graphs, Neuroquantology,20(19),(2022),426 - 435.
K.Lalithambigai and P.Gnanachandra, Topologies Induced on Vertex Set of
Graphs, Mapana-Journal of Sciences, 22(1),(2023),33 - 50.
C.Marijuan,Finite Topology and Digraphs,Proyecciones,29,(2010),291-307.
O. Njastad,Remarks on topologies defined by local properties,Avh.Norske
Vid.Akad Oslo I(N.S),8(1966),1-16
Shokry Nada, Abd El Fattah El Atik and Mohammed Atef,New Types of
Topological Structures via Graphs,Mathematical Methods in the Applied
Sciences,(2018),1-10.
R.Vaidyanathaswamy,Set Topology,Chelsea Publishing Company,(1960).
R.Vaidyanathaswamy,The Localization Theory in Set-topology,Proc.Indian. Acad.
Sci.20(1945),51-61.