Application of Semigraph and Directed Graph in Neurology of Mental
Illness (Anxiety)
1BIRGLAI BASUMATARY, 2,aABDUR ROHMAN, 3,b,*SURAJIT KR. NATH
1Department of Mathematical Sciences, Bodoland University, Kokrajhar, 783370, INDIA
2Department of Mathematical Sciences, Bodoland University, Kokrajhar, 783370, INDIA
3Department of Mathematical Sciences, Bodoland University, Kokrajhar, 783370, INDIA
ahttps://orcid.org/0009-0006-1353-6009, bhttps://orcid.org/0000-0002-2142-393X
*Corresponding author
Abstract— Semigraph is such a conception of a graph and it is similar to a graph when drawn in a plane. A
Semigraph contains more than two vertices. As the Graph Theory can abstract anything, the Semigraph can also
abstract anything which has any two edges at most one common vertex according to the definition. Besides in
the Medical field or human neurology, Semigraph can also be very useful.
The human body works with the functions of the brain and nervous system. The human body suffers
from some different types of diseases which may affect the human brain too or it may responsd to the diseases.
The research paper “Application of Semigraph and Directed Graph in Neurology of Mental Illness
(Anxiety)” represents how the semigraph and the directed graph can be used to know the diseases that affect
the human brain and how it may responsd to it.
Key-Words: - Semigraph, Directed Semigraph, Neurology, Human brain, Illness, Directed Graph
Received: March 21, 2024. Revised: April 11, 2024. Accepted: May 13, 2024. Published: June 10, 2024.
1 Introduction
The notion of semigraph, [1], is a generalization
of that of a graph. While generalizing a structure,
one naturally looks for one in which every concept
in the structure has a natural generalization.
Semigraph is such a natural generalization of a
graph and it resembles a graph when drawn in a
plane.
Semigraph is similar to a graph but it allows for
many new important or meaningful outcomes that
may or may not be possible with a Graph Theory,
[2]. The human brain functions similarly to a
computer’s CPU. It carries all the information
received from the human body senses, sends
messages back to the human body, and begins to
function.
As diseases affect the human body or the organs,
diseases can also affect the human brain which is
referred to as a mental disorder or a mental illness.
Most commonly the person suffers from anxiety
which affects the human brain and results in a
mentally depressed person.
It is crucial to know and comprehend quickly
how this sickness or condition might affect the
human brain and nervous system given that it is a
widespread disorder or illness among people
worldwide.
It’s kind of fascinating how the medical field
area interacts with Semigraph and Directed Graph
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2024.3.6
Birglai Basumatary, Abdur Rohman, Surajit Kr. Nath
E-ISSN: 2945-0454
74
Volume 3, 2024
Theory. Knowing what you could need in the future
is crucial in today’s society. To learn about a
condition or the relationship between an illness and
another in your neurology or any other body system,
you don’t have to work in the medical field.
The objectives of this work are, first create a
graphical representation of the disease analysis in
the medical field, then to apply Semigraph and
Directed Graph in modeling interactions between
objects, and finally to use Graph Theory to discover
the process of neurology of the Brain, [3], [4],
responding to Diseases.
2 Preliminary
Definitions
Graph: A graph H=(V,E) consist of a set of objects
H=[v1, v2,……..] called vertices and another set
E=[e1,e2............], whose elements are called edge.
Figure 1-Example of Graph
Directed Graph: A Directed Graph is made up of
an ordered collection of X pairs of distinct points
and a finite nonempty set V of points. Figure 2 is a
example for Directed Graph.
Figure 2-Example of Directed Graph
Semigraph: A semigraph G is a pair (V, X) where V
is a non-empty set whose elements are called
vertices of G and X is a set of n-tuples, called edges
of G, of distinct vertices, for various ,
satisfying the following conditions.
(a) Any two edges have at most one vertex in
common.
(b) Two edges (u1,u2,…,un) and (v1,v2,…,vm) are
considered to be equal if and only if
(i) m= n and
(ii) either ui = vi for ,
or ui = vn-i+1 for .
An edge e is represented by a simple open
Jordan curve which is drawn as a straight line whose
endpoints are called the end vertices of the edge e
and the m-vertices of the edge e each of which is not
an m-vertex of any other edge of the semigraph G
are denoted by small circles placed on the curve in
between the end vertices, in the order specified by e.
The end vertices of edges that are not m-vertices are
specially represented by thick dots. If an m-vertex of
an edge e is an end vertex of an edge e՛ i.e. an (m,
e) vertex, we draw a small tangent to the circle at
the end of the edge e՛.
Example 1: Let be a semigraph. Then
the edges of the semigraph in Figure 3 are
.
1
v
2
v
3
v
4
v
5
v
1
e
2
e
3
e
4
e
5
e
6
e
7
e
1
2
3
4
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2024.3.6
Birglai Basumatary, Abdur Rohman, Surajit Kr. Nath
E-ISSN: 2945-0454
75
Volume 3, 2024
Figure 3-Example of Semigraph
Dendroid: A Dendroid is a connected Semigraph
without strong cycles, and a forest is a semigraph in
which every component is a dendroid.
Pendant Dendroid: A Pendant Dendroid is a
dendroid with a pendant on each edge. For example
Figure 4 is a pendant dendroid
Figure 4-Example of Pendant Dendroid
3 Application of Semigraph in
Neurology of a Brain
3.1 Structure of a brain
In humans, the structure of a Brain Figure 5 which
has 100 billion of nerves system, controls every
aspect of function. Our emotions, including
happiness, tension, and depression as well as any
diseases are tied to or controlled by the brain, as I
have already discussed how the brain operates and
manages the human body similarly to how the CPU
manages a computer. Similar to anxiety, the
Amygdala is a component of the human brain that
has a neural system. The brain’s amygdala reacts to
any type of human emotion including stress and
worry.
Figure 5- Structure of a brain
Let sets of vertices in a Semigraph H be all
the name vertex (V, E). Where H=
{v1,v2,v3,v4,v5,v6,v7,v8,v9,v10} and the edges of
semigraph H are (v1,v2) , (v2, v3,v4) , (v1,v3), ( v1,v4).
This meets the requirements of the semigraph.
0
v
1
v
2
v
3
v
4
v
5
v
6
v
7
v
8
v
B
FB
MB
HB
TC
DC
CB
BG
HC
Any
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2024.3.6
Birglai Basumatary, Abdur Rohman, Surajit Kr. Nath
E-ISSN: 2945-0454
76
Volume 3, 2024
This demonstrates that, unlike the Brain structure
diagram below in Figure 6, which is complicated,
the structure of the area of the brain associated with
anxiety can be presented and illustrated simply.
Figure 6- Area of the brain
The major components of a brain’s anatomy are a
little more difficult for non-medical departments to
comprehend. Next observe Figure 7 and Figure 8 of
a Forebrain, which is divided into two portions, the
Telencephelon and Diencephelon, each of which has
a unique set of connecting structures and functions.
Instead of observing the other portions, let’s observe
the portion that contains the Amygdala, the portion
that reacts to anxiety.
Figure-7- Portion of Brain
Figure 8- Brain
3.2 Applications of Dendroid in Alzheimer’s
Disease:
The most prevalent kind of Dementia, Alzheimer’s
disease, primarily affects elderly people. Alois
Alzheimer in 1906 created the disease. Alzheimer’s
is a neurological disorder Figure 9 and Figure 10
that results in brain shrinkage and the loss of brain
cells.
Figure 9- Neurological disorder
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2024.3.6
Birglai Basumatary, Abdur Rohman, Surajit Kr. Nath
E-ISSN: 2945-0454
77
Volume 3, 2024
Figure 10- Alzheimer disease
Let us now discuss how Alzheimer’s disease
impacts the human brain using a Pendant Dendroid
Semigraph. Take a node of Figure 11.
Figure 11-Human brain using Pendant dendroid
The healthy brain is depicted in the first graph,
followed by Alzheimer’s disease’s attempt to
damage memory-related neurons, and the neurons
and the remainder of the vertex associated with
neurons, including the entorhinal and hippocampus,
are affected. The cerebral cortex is the last area
affected when it affects those three vertices
Figure 12- Pendent dendroid semigraph
The given Figure 12 is the pendant dendroid
semigraph which can be considered as a normal
brain of a healthy person. The triangle shapes of the
figure represent neurons of a healthy person’s brain,
assume it as v1. The vertex EC represents the
entorhinal cortex, consider it as v2, HC represents
the vertex v3 and at last, CC represents the vertex v4.
Figure 13-Alzheimer disease contact with neurons
Now, from Figure 13 consider a white vertex
entering the neurons, and assume it as an
Alzheimer’s disease which is trying to get contact
with Memory connected with a number of neurons.
EC
HC
CC
)(Neurons
healthy
person
EC
HC
CC
MN
EC
HC
CC
affected
EC
CC
HC
)(Neurons
healthy
person
EC
MN
CC
HC
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2024.3.6
Birglai Basumatary, Abdur Rohman, Surajit Kr. Nath
E-ISSN: 2945-0454
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Volume 3, 2024
Figure 14-Triangle shape of Alzheimer disease
From the above Figure 14 observe the triangle shape
colored completely which we can consider as totally
affected by Alzheimer’s disease.
In this way, we can figure out easily how
Alzheimer’s Disease can affect the parts of a healthy
person’s brain and know the way for the treatment
of a patient without observing to a critical structure
of a brain.
4 Directed Graph on Amygdala Responding to
Anxiety
The amygdala is the part of the human brain that has
a central role in anxiety responses to stress in whole
situations. During the response of the amygdala to
anxiety their activation influences anxiogenic
effects while their inactivation causes anxiolytic
effects. Also, neurotransmitter and stress mediators
having a role in amygdala nuclei can regulate the
character of anxiety.
All human beings go through anxiety in
some cases every person has faced it or facing it. It
is a physiological response that we face in our
everyday life. Anxiety is so common that it is one of
the most common symptoms of neurological
disorders.
In this section we will try to describe the
connections of neurons of a brain associated with
anxiety with the help of Directed Graph. It is a little
difficult to understand the process with the help of a
general diagram given below in Figure 4.1.
Figure 15-Directed graph of a brain
Now let us represent it by the directed graph,
observe the graph given below:
Figure 16 is the graph of a response of the
amygdala to anxiety when the patient is given an
anxiogenic projection.
EC
HC
CC
affected
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2024.3.6
Birglai Basumatary, Abdur Rohman, Surajit Kr. Nath
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Volume 3, 2024
Figure 16- anxiogenic projection
Here, the BLA represents a basolateral amygdala
considered as a vertex v1 which is directed to mPFC.
mPFC represents a medial prefrontal cortex. Again
HPC represents a Hippocampus directed to BLA.
And BLA, CEA, MeA are directed to BNST (Bed
nucleus of stria terminalise). And BNST is directed
to VTA (Ventral tegmental area) also to
adBNST(antidorsal BNST). Next let us observe the
other Figure 17 . Which determines the directed
graph, an Anxiolyti projection process of the
amygdala responding to Anxiety. Here the dot lines
represent the anxiolytic projection of the Amygdala
responding to anxiety where each named vertex is
directed to the other vertex creating an edge.
Figure 17- Anxiolyti projection process of the
amygdala responding to Anxiety
Next observe Figure 18, where each vertex is
directed to another vertex with an edge having a
transmitter process of the amygdala responding to
anxiety.
Figure 18- Transmitter process
Here BNST vertex is directed to LHA and VTA and
mPFC is directed to BLA and BLA is directed to
BNST.
BLA
CEA
MeA
PvN
LC
RN
BNST
HPC
mPFC
BLA
CEA
MeA
VTA
BNST
adBSNT
BLA
mPFC
VTA
BNST
LHA
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As mentioned all vertices (V) are the neurons
attached to the Amygdala and directed lines are the
edges (E) connecting to other vertex and it take part
during the process while the amygdala responds to
Anxiety.
5 Conclusion
In this research work, the we initiated application of
both Semigraph and Directed Graph Theory in the
Neurology of the human brain and also in a Disease
of a mental disorder or an illness with different
types of tools in different types of ways. We also
tried to show how the critical figure can be shown in
a simple and easy way to make understand in a
proper and easy way to the people with a Graph or a
Semigraph.
References:
[1] E. Sampathkumar, “Semigraph and Their
Applications”, Academy of Discrete Mathematics
and Applications, India, 2000
[2] F. Harary, “Graph Theory”, Addison-Wesley
Publishing Company, New York, 1969
[3] R. Carter, S.Aldridge, M. Page, “The Human
brain book”, 1st American ed. London, New
York, 2009
[4] S. Simon, “The brain: Our nervous system”,
Morrow Junior Books, New York, 1997
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
International Journal of Applied Sciences & Development
DOI: 10.37394/232029.2024.3.6
Birglai Basumatary, Abdur Rohman, Surajit Kr. Nath
E-ISSN: 2945-0454
81
Volume 3, 2024
