MATHEMATICAL MODELLING AND SIMULATION OF FINGERPRINT ANALYSIS USING GRAPH ISOMORPHISM, DOMINATION, AND GRAPH PEBBLING

In this work, an attempt has been made to represent and simulate fingerprint pattern analysis mathematically using graph isomorphism, graph dominance and graph pebbling. The idea of categorizing fingerprint by locating the ridge characteristics of fingerprint has been attempted to be implemented in this work. An image of fingerprint is used as an input for mathematical modelling and simulation. For the mathematical modelling and simulation of ridge characteristics in a fingerprint, we define two techniques. For first technique, we define Algorithm 1 A for finding the graphical representation of fingerprint and for second technique, we define Algorithm 2 A for determining whether the graphs of any two fingerprint patterns are isomorphic or not. The objective of this study is to identify that the two fingerprints are similar or different with the help of graph isomorphism, domination in graphs and graph pebbling. Our goal is to improve fingerprint identification accuracy in various orientations.


Introduction
The general method of converting a realistic problem into an abstract problem with a diagram and then analyzing and evaluating its characteristics is known as graph theory.It is used in the solution of many types of realistic problems arising in science, engineering and technology.Graph theory plays a vital role as a branch of discrete mathematics.A structural graph ( ) contains two sets where one is set of vertices and other one is set of edges joining certain pairs of these vertices.Mathematically, we can write ( ) ( ) [ ], , where ( ) ( ) 1 G E are two finite sets defined as: The proper attribution of evidence is a substantial challenge in the collection and classification of fingerprint evidence.While this can be a concern in any forensic investigation, forensic dermatoglyphics is particularly susceptible to it.Just linking a certain attack vector to a particular network or even computer is insufficient.The action in the problem must be linked to a specific user, according to the investigator.There is a chance that numerous people could access a single computer, especially in office networks.
The patterns present on the volar surfaces of the fingers are called fingerprints.The valleys between the friction ridges also known as skin ridges create distinctive patterns on the fingers.A biometric method called fingerprint analysis compares scanned images of prints to a database of fingerprints.Fingerprint analysis is based on the notion that each person's unique prints are constant throughout life.Because even the smallest changes in the local environment during fetal development contribute to the uniqueness of the prints, fingerprint analysis can distinguish between identical twins where even DNA fingerprinting cannot.The fingerprint pattern is unchanged, but the patterns have become larger due to growth.Moreover, certain diseases or accidents may distort fingerprint patterns.The characteristics that the ridges on the fingers produce are used to categorize and classify them.The most essential traits are ridge ends and bifurcations (separation of a ridge into two).These characteristics, known as ridge, serve as the foundation for additional classification and identification.As was said in the introduction, fingerprints are further divided into many more different patterns based on the shapes of the ridges of fingerprints.

Graph pebbling number
Graph pebbling is a mathematical game played on a graph with zero or more pebbles on each of its vertices.Game play is composed of a series of pebbling moves.Initially pebbles are placed on the vertices of a graph G according to a distribution D, a function ( ) is an operation of removing two pebbles from a vertex and adding one pebble to an adjacent vertex and that vertex of the graph must have at least two pebbles on it.Hence, a pebbling move can only be made on a vertex that has two or more pebbles.The least number 'm' is called a pebbling number of a vertex 'v' if m pebbles are randomly distributed over the graph G, then there must have some series of pebbling moves by which a pebble can be placed on vertex 'v'.The first step in applying graph theory to any mathematical modelling is to identify the various entities involved in the incident in that problem.

Graphical Representation of Fingerprint
Vertices will be used to represent these entities and edge between two vertices represents the relations between these vertices.The edge should always be an arc that represents both the initiator and the connector.Now our aim is to discuss a methodology for analyzing the fingerprints with the help of graph isomorphism, domination and graph pebbling.So, for describing this methodology, first we have to construct the graphical network of fingerprints.Therefore, throughout this study for finding the vertices and edges of graphical network of fingerprints, we use some special vocabulary for the ridge characteristics as follows: A by assuming ridge characteristics as the vertices of graphical representation.

Algorithm 1 A
Step I. Determine all the ridge characteristics in the fingerprint as discussed in Table 3.1 such that they should cover as much of the print as practicable for that fingerprint.
Step II.Associate a vertex to each of the ridge characteristics in the corresponding fingerprint.
Step III.Draw an edge between any two vertices that lie on a "connected ridge flow." Step IV.Use different colors for different types of ridge characteristics (for example: green for Bifurcation (B), red for Ridge Ending (RE), yellow for Ridge Crossing (RC), blue for Double Bifurcation (DB), etc.).
Step V. Associate each vertex of graph as a pair of numbers ( ) n m, of the ridge characteristics in the corresponding fingerprint as follows: • First component of pair ( ) = m the corresponding abbreviation of the ridge characteristic associated with the vertex.
• Second component of pair ( ) = n the degree of the vertex.

Illustration
We have a fingerprint 1 F as shown in Figure 3. Now, we find the graphical representation of this fingerprint with the help of Algorithm .Now, for finding the graphical representation, first we have to identify all the ridge characteristics in the fingerprint such that they should cover as much of the print as practicable for that fingerprint.After identifying all the ridge characteristics in the fingerprint, define each ridge characteristic by special vocabulary as we discussed earlier.Associate a vertex (ridge characteristics) to each of the ridge characteristics and make a relation between any two vertices that lies on the same connected components.Therefore, we get the graphical representation of given fingerprint as shown in Figure 4.

Methodology for Identification of Fingerprints
There are two main subcategories of contemporary graph theory.The first deals with graph theory's algebraic components.The representation of the vertices and edges with the weight is the main concerns in this subtopic of graph theory.In essence, this uses graph theory as a descriptive application.Graph theory's second subfield focuses on optimization issues.
The path of data over a network has been optimized using this feature of graph theory in network traffic analysis.Algebraic graph theory is the main topic of this essay.Nonetheless, it is undoubtedly conceivable to use graph theory's optimization features for the examination of forensic data.To build on the concepts presented in this study, it is advised that optimization-related parts of graph theory will be taken into account as potential research topics in the future.There are numerous forensic science applications for graph theory.The present methodology is focused on the identification of fingerprints and assesses the relationships between specific pieces of evidence, suspects, victims and any other entities pertinent to a given inquiry.

Probability distribution matrix
Let G be any strongly connected directed graph of any fingerprint with adjacency matrix [ ].
Then the probability distribution matrix of that fingerprint is defined by where i d is the degree of ith vertex of graph such that the direction is taken from vertex i to vertex j.

Probability propagation matrix
We define probability propagation matrix by defining a probability finite automaton (PFA) U in form of four tuples ( ) • ε is defined as the domain of transition function.
• M is described as a transition function whose domain and codomain are ε and n n × matrices, respectively, where is the probability for which PFA U exists and U changes from state i to state j after determining that the symbol is stochastic, i.e., for any state i,

,
• The last tuple ( ) Now with an initial state ( ) 1 and reading string x, the state distribution vector for the PFA U is defined as ( ) ( ).
Now, we calculate the related PPM matrix [ ] by finding all the state distribution vectors ( ),

we use an algorithm 2
A to determine whether the fingerprint 1 F is similar to the fingerprint .

Algorithm 2 A
Step I. Find the graph of both the given fingerprints as 1 G and . 2

G
Step II.Convert both the graphs 1 G and 2 G to the strongly connected diagraphs ( ) Step III.Find the adjacency matrix of both the strongly connected diagraphs and convert 1 G and 2 G to the corresponding probability finite automaton (PFA) 1 U and , 2 U respectively.
Step IV.Calculate the related probability distribution matrices (PDM) A and .

2
A Using the state distribution vectors defined above, discover the related PPM matrices ( ) ( ).G and 2 G of fingerprints 1 F and . 2

F
Step VI. ( ) ( ) are isomorphic to each other if and only if for any fixed number i, there exist some j, n j ≤ ≤ 1 such that their probability propagation matrices i P 1 and i P 2 are isomorphic.

Mathematical Modelling and Simulation of Fingerprint Analysis
In this section, we focus on the application part of Algorithms 1 A and .

2
A By utilizing algorithms, we model and simulate fingerprint analysis mathematically.

Experiment 1
In Experiment 1, we take any two random images of fingerprints 1 F and 2 F as shown in Figure 5.By applying Algorithm , A we find graph of both the fingerprints and then, we apply Algorithm 2 A for analyzing that both the fingerprints are similar or not. 2 The adjacency matrix for the graph 1 G is ( ).
The initial state distribution vector 1 γ is now obtained by assuming that the PFA 2nd state is used as the initial state.So, . 1 The probability distribution vectors ( ), 2  a a of PFA are now as follows with regard to the starting state distribution vector: Now, we find the probability propagation matrix of graph 2 G of fingerprint . 2 The adjacency matrix for the graph 2 G is .The probability distribution matrix for the graph 2 G is ( ).
The initial state distribution vector 2 γ is now obtained by assuming that the PFA 2nd state is used as the initial state.So, The probability distribution vectors ( ), U are now as follows with regard to the starting state distribution vector: So, finally we have the probability propagation matrices ( )
in the neighborhood of some vertex in set D and the dominating number ( ) G γ of any G is the least number of members in a dominating set of graph G [5].

F
Find the domination number and graph pebbling number of both the graphs 1

Table 3 .1 BR
Mathematical Modelling and Simulation of Fingerprint … 283 We analyze that the domination number and graph pebbling number of both the graphs are same.Hence, by Algorithm , In the present investigation, we utilized graph isomorphism, domination in graphs, and graph pebbling to model and simulate mathematically fingerprints to see if the two fingerprints are identical or different.For the mathematical modelling and simulation of fingerprint analysis, we created two methods.The first Algorithm 1 A finds a fingerprint's graphical representation, while the second Algorithm 2 A determines whether or not any two fingerprints' graphs are isomorphic.Our objective is to increase the accuracy of fingerprint identification in diverse orientations.This will help the investigators to make comparable fingerprint analysis more efficiently in less time.