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A COMPREHENSIVE STUDY OF VARIOUS APPLICATION OF IN MODELING: PROSPECTIVE OF VARIOUS GRAPH COLORING AND DIRECTIONS

1K.VARALAXMI, RAPOLU MANJULA2, GATTU PRASAD3, K.SANTHOSH KUMAR4

1Asst.prof,Holymary institute of technology and science, Keesara - Bogaram - Ghatkesar Rd, Kondapur, Telangana 501301 234Asst.prof ,Vignan's Institute of management and technology for women, Kondapur, Ghatkesar, Ranga Reddy, Telangana

Abstract: Graph theory is quickly moving into the standard of arithmetic for the most part in light of its applications in various fields which incorporate organic chemistry (genomics), electrical building (interchanges systems and coding theory), software engineering (calculations and calculations) and tasks inquire about (booking). The incredible combinatorial strategies found in graph theory have likewise been utilized to demonstrate major outcomes in different zones of unadulterated arithmetic. Graphs are utilized to characterize the progression of calculation. Graphs are utilized to speak to systems of correspondence. Graphs are utilized to speak to information association. Graph change frameworks take a shot at rule-situated in-memory control of graphs. In arithmetic, graph theory is the investigation of graphs, which are numerical structures, used to display pairwise relations between objects. A graph right now made up of vertices (additionally called hubs or focuses) which are associated by edges (likewise called connections or lines). The paper along these lines centers around the various parts of this incredible technique for representation of logical realities that can be utilized to tackle some constant problems. The accompanying paper presents the peruser with the presentation, phrasing of graph theory. Imminent of different Graph Coloring and directions applications of graph theory in the assorted fields of science and innovation. Keywords: Coloring Applications, Graph Coloring, Graph Labeling, Modeling, Problem Solving Techniques, Representation. I. INTRODUCTION In , graph theory is the investigation of graphs, which are scientific structures used to show pairwise relations between objects. A graph right now made up of vertices (likewise called hubs or focuses) which are associated by edges (additionally called connections or lines). A differentiation is made between undirected graphs, where edges interface two vertices evenly and coordinated graphs, where edges connect two vertices unevenly; see Graph (discrete science) for increasing point by point definitions and for different varieties in the sorts of the graph that are normally thought of. Graphs are one of the prime objects of study in discrete science.

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Over one century after Euler's paper on the extensions of Königsberg and keeping in mind that Listing was presenting the idea of topology, Cayley was driven by an enthusiasm for specific expository structures emerging from differential analytics to contemplate a specific class of graphs, the trees.[22] This examination had numerous ramifications for hypothetical science. The techniques he utilized fundamentally concern the list of graphs with specific properties. Enumerative graph theory at that point emerged from the aftereffects of Cayley and the essential outcomes distributed by Pólya somewhere in the range of 1935 and 1937. These were summed up by De Bruijn in 1959. Cayley connected his outcomes on trees with contemporary investigations of synthetic composition.[23] The combination of thoughts from science with those from science started what has become some portion of the standard phrasing of graph theory. Specifically, the expression "graph" was presented by Sylvester in a paper distributed in 1878 in Nature, where he draws a similarity between "quantic invariants" and "co-variations" of polynomial math and atomic diagrams:[24] The primary reading material on graph theory was composed by Dénes Kőnig and distributed in 1936.[25] Another book by Frank Harary, distributed in 1969, was "considered the world over to be the complete course reading on the subject",[26] and empowered mathematicians, physicists, electrical architects, and social researchers to converse with one another. Harary gave the entirety of the sovereignties to subsidize the Pólya Prize.[27] One of the most popular and animating problems in graph theory is the four-shading problem: "Is it genuine that any guide attracted the plane may have its districts hued with four hues so that any two locales having a typical outskirt have various hues?" This problem was first presented by Francis Guthrie in 1852 and its originally set up account is in a letter of De Morgan routed to Hamilton that year. Numerous wrong evidence has been proposed, including those by Cayley, Kempe, and others. The investigation and the speculation of this problem by Tait, Heawood, Ramsey, and Hadwiger prompted the investigation of the colorings of the graphs inserted on surfaces with the discretionary class. Tait's reformulation created another class of problems, the factorization problems, especially concentrated by Petersen and Kőnig. Crafted by Ramsey on hues and all the more particularly the outcomes acquired by Turán in 1941 were at the cause of another part of graph theory, extremal graph theory. The four-shading problem stayed unsolved for over a century. In 1969 Heinrich Heesch distributed a strategy for solving the problem of utilizing computers.[28] A PC supported verification delivered in 1976 by Kenneth Appel and Wolfgang Haken utilizes the thought of "releasing" created by Heesch.[29][30] The confirmation included checking the properties of 1,936 designs by PC and was not completely acknowledged at the time because of its multifaceted nature. A less complex verification considering just 633 arrangements was given twenty years after the fact by Robertson, Seymour, Sanders, and Thomas.[1] The self-sufficient advancement of topology from 1860 and 1930 prepared graph theory back through crafted by Jordan, Kuratowski and Whitney. Another significant factor of normal

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improvement of graph theory and topology originated from the utilization of the techniques of current polynomial math. The principal case of such use originates from crafted by the physicist Gustav Kirchhoff, who distributed in 1845 his Kirchhoff's circuit laws for figuring the voltage and flow in electric circuits. The presentation of probabilistic strategies in graph theory, particularly in the investigation of Erdős and Rényi of the asymptotic likelihood of graph network, offered to ascend to one more branch, known as arbitrary graph theory, which has been a productive wellspring of graph- theoretic outcomes. Graph drawing Graphs are spoken to outwardly by drawing a point or hover for each vertex and drawing a line between two vertices on the off chance that they are associated with an edge. In the event that the graph is coordinated, the bearing is demonstrated by drawing a bolt. A graph drawing ought not to be mistaken for the graph itself (the theoretical, non-visual structure) as there are a few different ways to structure the graph drawing. The only thing that is important is which vertices are associated with which others by what number of edges and not the specific design. By and by, it is frequently hard to choose if two drawings speak to a similar graph. Contingent upon the problem space a few designs might be more qualified and more obvious than others. The spearheading work of W. T. Tutte was powerful regarding the matter of graph drawing. Among different accomplishments, he presented the utilization of direct mathematical techniques to acquire graph drawings. Graph drawing additionally can be said to incorporate problems that manage the intersection number and its different speculations. The intersection number of a graph is the base number of crossing points between edges that a drawing of the graph in the plane must contain. For a planar graph, the intersection number is zero by definition.

II. GRAPH THEORY APPLICATIONS

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The system graph framed by Wikipedia editors (edges) adding to various Wikipedia language forms (vertices) for one month in summer 2013.[6] Graphs can be utilized to demonstrate numerous sorts of relations and procedures in physical, biological,[7][8] social and data frameworks. Numerous functional problems can be spoken to by graphs. Accentuating their application to true frameworks, the term arrange is at times characterized to mean a graph in which properties (for example names) are related with the vertices and edges, and the subject that communicates and comprehends this present reality frameworks as a system is called arrange science. Software engineering In software engineering, graphs are utilized to speak to systems of correspondence, information association, computational gadgets, the progression of calculation, and so on. For example, the connection structure of a site can be spoken to by a coordinated graph, wherein the vertices speak to pages and coordinated edges speak to joins starting with one page then onto the next. A comparative methodology can be taken to problems in social media,[9] travel, science, PC chip configuration, mapping the movement of neurodegenerative diseases,[10][11] and numerous different fields. The advancement of calculations to deal with graphs is consequently of significant enthusiasm for software engineering. The change of graphs is regularly formalized and spoken to by graph revise frameworks. Integral to graph change frameworks concentrating on rule-situated in-memory control of graphs are graph databases outfitted towards exchange sheltered, relentless putting away and questioning of graph-organized information. Phonetics Graph-theoretic techniques, in different structures, have demonstrated especially valuable in phonetics, since characteristic language regularly loans itself well to discrete structure. Generally, sentence structure and compositional semantics follow tree-based structures, whose expressive force lies in the standard of compositionality, displayed in a various leveled graph. Progressively contemporary methodologies, for example, head-driven expression structure sentence structure model the grammar of regular language utilizing composed component structures, which are coordinated non-cyclic graphs. Inside lexical semantics, particularly as applied to PCs, modeling word significance is simpler when a given word is comprehended as far as related words; semantic systems are in this manner significant in computational etymology. All things considered, different strategies in phonology (for example optimality theory, which utilizes grid graphs) and morphology (for example limited state morphology, utilizing limited state transducers) are basic in the investigation of language as a graph. For sure, the handiness of this territory of science to etymology has borne associations, for example, TextGraphs, just as different 'Net' ventures, for example, WordNet, VerbNet, and others.

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Material science and science Graph theory is likewise used to contemplate atoms in science and material science. In dense issue material science, the three-dimensional structure of confounded reproduced nuclear structures can be concentrated quantitatively by get-together measurements on graph-theoretic properties identified with the topology of the particles. Likewise, "the Feynman graphs and rules of estimation abridge quantum field theory in a structure in close contact with the exploratory numbers one needs to understand."[12] In science a graph makes a characteristic model for a particle, where vertices speak to iotas and edges bonds. This methodology is particularly utilized in PC preparing of sub-atomic structures, running from concoction editors to database looking. In measurable material science, graphs can speak to neighborhood associations between interfacing parts of a framework, just as the elements of a physical procedure on such frameworks. So also, in computational neuroscience graphs can be utilized to speak to utilitarian associations between cerebrum zones that cooperate to offer ascent to different intellectual procedures, where the vertices speak to various territories of the mind and the edges speak to the associations between those zones. Graph theory assumes a significant job in electrical modeling of electrical systems, here, loads are related with opposition of the wire portions to get electrical properties of system structures. Graphs are additionally used to speak to the miniaturized scale channels of permeable media, in which the vertices speak to the pores and the edges speak to the littler channels associating the pores. Compound graph theory utilizes the sub-atomic graph as a way to display particles. Graphs and systems are incredible models to contemplate and comprehend stage changes and basic marvels. Expulsion of hubs or edges prompts a basic progress where the system breaks into little bunches which are concentrated as a stage change. This breakdown is considered by means of the permeation theory.[14] [15]

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Sociologies

Graph theory in human science: Moreno Sociogram (1953).[16] Graph theory is likewise generally utilized in humanism as away, for instance, to gauge on-screen characters' distinction or to investigate talk spreading, strikingly using informal community examination programming. Under the umbrella of informal communities are various sorts of graphs.[17] Acquaintanceship and kinship graphs portray whether individuals know one another. Impact graphs model whether certain individuals can impact the conduct of others. At last, the cooperation graphs model whether two individuals cooperate with a certain goal in mind, for example, acting in a film together.

Science

In like manner, graph theory is helpful in science and protection endeavors where a vertex can speak to districts where certain species exist (or occupy) and the edges speak to relocation ways or development between the locales. This data is significant when seeing reproducing examples or following the spread of illness, parasites or how changes to the development can influence different species. Graph theory is additionally utilized in connectomics;[18] sensory systems can be viewed as a graph, where the hubs are neurons and the edges are the associations between them.

Arithmetic

In arithmetic, graphs are helpful in geometry and certain pieces of topology, for example, tie theory. Arithmetical graph theory has close connections with bunch theory. Logarithmic graph theory has been applied to numerous zones including dynamic frameworks and intricacy.Social sciences

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III. GRAPH COLORING In graph theory, graph coloring is an uncommon instance of graph labeling; it is a task of names generally called "hues" to components of a graph subject to specific requirements. In its least difficult structure, it is a method for coloring the vertices of a graph with the end goal that no two neighboring vertices are of a similar shading; this is known as a vertex coloring. Additionally, an doles out a shading to each edge with the goal that no two nearby edges are of a similar shading, and a face coloring of a planar graph relegates a shading to each face or area so no two faces that share a limit have a similar shading.

Vertex coloring is generally used to present graph coloring problems since other coloring problems can be changed into a vertex coloring example. For instance, an edge coloring of a graph is only a vertex coloring of its , and a face coloring of a plane graph is only a vertex coloring of its double. In any case, non-vertex coloring problems are regularly expressed and concentrated with no guarantees. This is halfway educational, and incompletely on the grounds that a few problems are best concentrated in their non-vertex structure, as on account of edge coloring.

The show of utilizing hues starts from coloring the nations of a guide, where each face is actually shaded. This was summed up to coloring the essences of a graph inserted in the plane. By planar duality it became coloring the vertices, and right now sums up to all graphs. In numerical and PC representations, it is common to utilize the initial hardly any positive or non-negative whole numbers as the "hues". By and large, one can utilize any limited set as the "shading set". The idea of the coloring problem relies upon the quantity of hues yet not on what they are.

Graph coloring appreciates numerous pragmatic applications just as hypothetical difficulties. Adjacent to the old style kinds of problems, various impediments can likewise be determined to the graph, or in transit a shading is alloted, or even on the shading itself. It has even arrived at fame with the overall population as the well known number riddle Sudoku. Graph coloring is as

yet an exceptionally dynamic field of research.

An appropriate vertex coloring of the with 3 hues, the base number conceivable.

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Vertex coloring At the point when utilized with no capability, a coloring of a graph is quite often an appropriate vertex coloring, in particular a labeling of the graph's vertices with hues to such an extent that no two vertices having a similar edge have a similar shading. Since a vertex with a circle (for example an association legitimately back to itself) would never be appropriately shaded, it is comprehended that graphs right now loopless. The wording of utilizing hues for vertex names returns to delineate. Names like red and blue are possibly utilized when the quantity of hues is little, and regularly it is comprehended that the marks are drawn from the numbers {1, 2, 3, ...}.

A coloring utilizing all things considered k hues is known as an (appropriate) k-coloring. The most modest number of hues expected to shading a graph G is called its chromatic number, and is frequently meant χ(G). Here and there γ(G) is utilized, since χ(G) is additionally used to signify the Euler normal for a graph. A graph that can be appointed an (appropriate) k-coloring is k-colorable, and it is k-chromatic if its chromatic number is actually k. A subset of vertices relegated to a similar shading is known as a shading class, each such class frames a free set. In this manner, a k-coloring is equivalent to a segment of the vertex set into k free sets, and the terms k-partite and k-colorable have a similar importance. Chromatic polynomial All non-isomorphic graphs on 3 vertices and their chromatic polynomials. The vacant graph E3 (red) concedes a 1-coloring, the others concede no such colorings. The green graph concedes 12

colorings with 3 hues.

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The chromatic polynomial tallies the quantity of ways a graph can be hued utilizing close to a given number of hues. For instance, utilizing three hues, the graph in the adjoining picture can be shaded in 12 different ways. With just two hues, it can't be shaded by any means. With four hues, it very well may be hued in 24 + 4⋅12 = 72 different ways: utilizing every one of the four hues, there are 4! = 24 legitimate colorings (each task of four hues to any 4-vertex graph is an appropriate coloring); and for each decision of three of the four hues, there are 12 substantial 3- colorings. The chromatic polynomial incorporates at any rate as much data about the colorability of G as does the chromatic number. For sure, χ is the littlest positive whole number that isn't a base of the chromatic polynomial Edge coloring An edge coloring of a graph is a legitimate coloring of the edges, which means a task of hues to edges with the goal that no vertex is occurrence to two edges of a similar shading. An edge coloring with k hues is known as a k-edge-coloring and is identical to the problem of dividing the edge set into k matchings. The most modest number of hues required for an edge coloring of a graph G is the chromatic list, or edge chromatic number, χ′(G). A Tait coloring is a 3-edge coloring of a cubic graph. The four shading hypothesis is identical to the affirmation that each planar cubic bridgeless graph concedes a Tait coloring. Complete coloring Complete coloring is a kind of coloring on the vertices and edges of a graph. At the point when utilized with no capability, an all out coloring is constantly thought to be appropriate as in no adjoining vertices, no nearby edges, and no edge and its end-vertices are allocated a similar shading. The complete chromatic number χ″(G) of a graph G is the least hues required in any all out coloring of G. Unlabeled coloring An unlabeled coloring of a graph is a circle of a coloring under the activity of the automorphism gathering of the graph. In the event that we decipher a coloring of a graph on vertices as a vector in , the activity of an automorphism is a stage of the coefficients of the coloring. There are analogs of the chromatic polynomials which check the quantity of unlabeled colorings of a graph from a given limited shading set.

IV. VARIOUS GRAPH DIRECTIONS IN GRAPH THEORY In arithmetic, and all the more explicitly in graph theory, a coordinated graph (or digraph) is a graph that is comprised of a lot of vertices associated by edges, where the edges have a bearing related with them. Ubclasses

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A simple

A tournament on 4 vertices

A straightforward coordinated non-cyclic graph A competition on 4 vertices

• Symmetric coordinated graphs are coordinated graphs where all edges are bidirected (that is, for each bolt that has a place with the digraph, the comparing inversed bolt additionally has a place with it).

• Simple coordinated graphs are coordinated graphs that have no circles (bolts that legitimately interface vertices to themselves) and no different bolts with a similar source and target hubs. As of now presented, on account of various bolts the element is normally tended to as a coordinated multigraph. A few creators portray digraphs with circles as circle digraphs.[2]

o Complete coordinated graphs are straightforward coordinated graphs where each pair of vertices is joined by a symmetric pair of coordinated bolts (it is comparable to an undirected with the edges supplanted by sets of converse bolts). It follows that a total digraph is symmetric.

o Oriented graphs are coordinated graphs having no bidirected edges (for example at generally one of (x, y) and (y, x) might be bolts of the graph). It follows that a coordinated graph is a situated graph if and just on the off chance that it hasn't any 2-cycle.[3]

Tournaments are situated graphs acquired by picking a heading for each edge in undirected complete graphs. Directed non-cyclic graphs (DAGs) are coordinated graphs with no coordinated cycles.

Multitrees are DAGs in which no two coordinated ways from a solitary beginning vertex meet back at a similar closure vertex.

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Oriented trees or are DAGs shaped by arranging the edges of undirected non- cyclic graphs.

Rooted trees are situated trees in which all edges of the hidden undirected tree are coordinated either away from or towards the root.

Digraphs with valuable properties

This rundown is deficient; you can help by extending it.

• Weighted coordinated graphs (otherwise called coordinated systems) are (straightforward) coordinated graphs with loads alloted to their bolts, correspondingly to weighted graphs (which are otherwise called undirected systems or weighted networks).[2]

o Flow systems are weighted coordinated graphs where two hubs are recognized, a source and a sink.

• Rooted coordinated graphs (otherwise called stream graphs) are digraphs in which a vertex has been recognized as the root.

o Control stream graphs are established digraphs utilized in software engineering as a representation of the ways that may be navigated through a program during its execution.

• Signal-stream graphs are coordinated graphs in which hubs speak to framework factors and branches (edges, bends, or bolts) speak to practical associations between sets of hubs.

• Flow graphs are digraphs related with a lot of straight logarithmic or differential conditions.

• State graphs are coordinated multigraphs that speak to limited state machines.

• Commutative graphs are digraphs utilized in class theory, where the vertices speak to (scientific) objects and the bolts speak to morphisms, with the property that every single coordinated way with a similar beginning and endpoints lead to a similar outcome by arrangement.

• In the theory of Lie gatherings, a quiver Q is a coordinated graph filling in as the area of, and in this way describing the state of, a representation V characterized as a functor, explicitly an object of the functor class FinVctKF(Q) where F(Q) is the free classification on Q comprising of ways in Q and FinVctK is the classification of limited dimensional vector spaces over a field K.

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Representations of a quiver name its vertices with vector spaces and its edges (and henceforth ways) perfectly with straight changes among them, and change through normal changes.

Fundamental wording

Situated graph with comparing rate framework

A bolt (x, y) is viewed as guided from x to y; y is known as the head and x is known as the tail of the bolt; y is said to be an immediate successor of x and x is said to be an immediate forerunner of y. On the off chance that a way leads from x to y, at that point y is said to be a successor of x and reachable from x, and x is said to be a forerunner of y. The bolt (y, x) is known as the rearranged bolt of (x, y).

The nearness grid of a multidigraph with circles is the whole number esteemed framework with lines and segments relating to the vertices, where a nondiagonal passage aij is the quantity of bolts from vertex I to vertex j, and the corner to corner section aii is the quantity of circles at vertex I. The contiguousness network of a coordinated graph is extraordinary up to indistinguishable change of lines and segments.

Another framework representation for a coordinated graph is its frequency grid.

See the bearing for additional definitions.

Indegree and outdegree

A coordinated graph with vertices named (indegree, outdegree)

For a vertex, the quantity of head closes nearby a vertex is known as the indegree of the vertex and the quantity of the last parts adjoining a vertex is its outdegree (called "stretching factor" in trees).

Let G = (V, An) and v∈V. The in-level of v is indicated deg−(v) and its outdegree is signified deg+(v).

A vertex with deg−(v) = 0 is known as a source, as it is the cause of every one of its outcoming bolts. Likewise, a vertex with deg+(v) = 0 is known as a sink, since it is the finish of every one of its approaching bolts.

In the event that a vertex is neither a source nor a sink, it is called an internal.[citation needed]

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The degree whole equation expresses that, for a coordinated graph,

On the off chance that for each vertex v∈V, deg+(v) = deg−(v), the graph is known as a decent coordinated graph.[4]

Degree succession

The degree succession of a coordinated graph is the rundown of its in-degree and outdegree sets; for the above model we have degree arrangement ((2, 0), (2, 2), (0, 2), (1, 1)). The degree succession is a coordinated graph invariant so isomorphic coordinated graphs have a similar degree arrangement. Nonetheless, the degree grouping doesn't, by and large, extraordinarily recognize a coordinated graph; at times, non-isomorphic digraphs have a similar degree succession.

CONCLUSION

Graph theory is a settled, scientific technique for assessing connections. Right now applications of graph theory are appeared with models. Specifically, the idea of vertex availability and edge network of a graph can have a wide range of genuine applications and are additionally utilized as graph-theoretic instruments to examine the different problems at a convergence. Broad just as serious utilization of graph theory including the techniques of graph labeling and graph coloring to speak to and besides take care of problems by making legitimate problem-solving techniques are a significant device in many significant fields of fundamental sciences and current innovation. The whole graph, including vertices, circular segments, and rate capacities must be assessed so as to have a precise model of the examination. These fields incorporate yet are not constrained to different system the board zones, for example, transportation systems, strategic systems, activities systems, remote systems, and natural systems. Graph labeling is additionally an exceptionally compelling device in solving problems in genomics, material science, science, science, hereditary qualities and PC and media transmission applications.

REFERENCE:

1. Hubert, L. J. (1974). Some applications of graph theory to clustering. Psychometrika, 39(3), 283-309. 2. Harary, F., & Norman, R. Z. (1953). Graph theory as a mathematical model in social science (p. 45). Ann Arbor: , Institute for Social Research. 3. Gross, J. L., & Yellen, J. (2005). Graph theory and its applications. CRC press. 4. Bunn, A. G., Urban, D. L., & Keitt, T. H. (2000). Landscape connectivity: a conservation application of graph theory. Journal of environmental management, 59(4), 265-278. 5. Gross, J. L., & Yellen, J. (2005). Graph theory and its applications. CRC press.

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6. Fleurent, C., & Ferland, J. A. (1996). Genetic and hybrid algorithms for graph coloring. Annals of Operations Research, 63(3), 437-461. 7. Tuza, Z. (1997). Graph coloring with local constraints-A survey. 8. Roberts, F. S. (1978). Graph theory and its applications to problems of society (Vol. 29). SIAM. 9. L. Caccetta and K. Vijayan, “Applications of graph theory”, Fourteenth Australasian Conference on Combinational Mathematics and Computing, Dunedin, 1986, Ars. Combin. Vol. 23-1987, 21-27. 10. K. Heinrich and P. Horak, “Euler’s Theorem”, Am. Math. Monthly, 1994, Vol. 101,260. 11. E. Burman and P. Horak, “Some applications of graph theory to other Parts of Mathematics”, The Mathematical Intelligencer, Springer-Velag, , 1996, 6-11. 12. S. Wadhwa, “Analysis of a Network Design Problem”, Lehigh Universty, 2000. 13. Amaral, L. A., & Ottino, J. M. (2004). Complex networks. The European Physical Journal B- Condensed Matter and Complex 14. Systems, 38(2), 147-162. 15. [8]. Zufferey, A., Ratle, F., Ribaud, O., Esseiva, P, & Kanevski, M. (2006). Pattern Detection in Forensic Case Data Using Graph 16. Theory: Application to Heroin Cutting Agents. Forensic Science International 167 (2-3), pp 242–246. 17. Haggerty, J., Karran, A., Lamb, D., & Taylor, M. (2011). A Framework for the Forensic Investigation of Unstructured EmailRelationship Data. International Journal of Digital Crime and Forensics, 3(3), 1-18. 18. Gibilisco, S. (2004). Statistics Demystified. , NY: McGraw-Hil McKendall, Marie A., and John A. Wagner III. "Motive, opportunity, choice, and corporate illegality." Organization Science 8.6(1997): 624-647. 19. Godsil, C., & Royle, G. F. (2013). Algebraic graph theory (Vol. 207). Springer Science & Business Media 20. Trudeau, R. (1994). Introduction to Graph Theory. Mineola, New York: Dover Publications 21. Bondy, A., Murty, U. (2008). Graph Theory. New York City, NY: Springer Publishing

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