Application Of Graph Theory

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2. Graph theory, branch of mi

3. ES m
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thinking about ie following problem. Königsberg is divid in four pars by the
river Preg connected by seve! Ist possible to tour Königsberg along
la path that crosses every bridge once, and at most once? You can start and finish
wherever you want, not necessarily in the same place

Trying various paths very quickly suggests that it is impossible: you
lalways end up with a bridge you haven't crossed, but with no way to get there
without crossing another bridge twice, But it would take a very long time to.

try every possible path,

Having solved the Drawing Graph problem, we can now solve the
Königsberg Bridges problem by converting map of Königsberg into graph, Every
island is represented by a vertex, and every bridge connecting two islands is
represented by an edge connecting the two points.Here is what w

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A road network can be considered as a graph with positive wei

The nodes represent road junctions and each edge ofthe graph is associated with af

Load segment between two junctions. The we dge may correspond to th

length of the associated road segment, the ti the segment orth
M of traversing the segment, Using directed e sible to m

ay streets, Such graphs are special in the se

khan others for long distance travel ( ays). This property has b

formalized using the notion of highway dimension. There are a great numbe
orthms that exploit this property and are the

path a lot quicker than would be possible on g

All of th ithms work in two phases. Inthe first phase, th
lis preprocessed without knowing the source or target node. This phase may tal
several days for realistic data and some techniques. The second phase is the quer
phase. In this phase, source and target node are known. The running time of the
second phase is generally less than a second, The idea is that the road network
static, so the preprocessing phase can be done once and used for a large number of
[queries on the same road network

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Chemical graph theory is the topology branch of mathematical
[chemistry which applies graph theory to mathematical modelling of chemical
[phenomena. The pioneers of the chemical graph theory areAlexandru Balaban, Ante
[Graovac, Ivan Gutman, Haruo Hosoya, Milan Randié and Nenad Trinajs

(also Harry Wiener and others). In 1988, it was reported that several hundred
researchers worked in this area producing about 500 articles annually. A number of
Imonographs have been written in the area, including the two-volume comprehensive
ext by Trinajstic, Chemical Graph Theory, that summarized the field up to mid-
1980s

q The adherents of the theory maintain that the properties of a chemie
graph ic. a graph-theoretical representation of a molecule gives valuable insights,

lito the chemical phenomena. The opponents contend that graphs play only a fringe

le in chemical research, One variant of the theory is the representation of materials

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In chemical graph theory and in mathematical chemistry, a molecular
[graph or chemical graph is a representation of the structural formula of a chemical
[compound in terms of graph theory. A chemical graph is a labeled
10 the atoms of the compou es correspond t
inds of the corresponding atoms and

abeled with the types of bonds. For purposes any of the lab

be ignored.
A hydro

[molecular graph is the molecul:

Molecular graphs can disti

[compounds which have the same molecular formula but non-isomorphic graphs =

[such as iso-pentane and neo-pentane. On the other hand, the molecular g

Inormally does not contain any information about the three-din

lor the bonds, and therefore cannot distinguish between conformational isomers (such

las cis and trans 2-butene) or stereoisomer (such as D- and L-glyceraldehydes),

q In some important cases (topological index calculation ete.) the

following classical definition is sufficient: molecular graph is connected undirected
aph one-to-one corresponded to structural formula of chemical compound so that

Ivertices of the graph correspond to atoms of the molecule and edges of the graph

[correspond to chemical bond n these atoms, One variant is to represent

rials as infinite Euclidean graphs, in particular, crystals as periodic graphs.

13. periodic graph or crystal net is a three:

mal periodic graph, Le. a three-dimensional Euclidea

points in three-

5 of atoms sı
building units, whi

al Euclidean space, and

22. Graph-iheoretic methods, in various forms, have proven particularly
useful in linguistics, since natural language often lends itself well to discrete
structure, Traditionally, syntax and compositional semantis follow tree-based
structures, whose expressive power lies in the principle of compositionality, modeled
in a hierarchical graph. More contemporary approaches such as head-driven phrase
structure grammar model the syntax of natural language using typed feature
structures, which are directed acyclic graphs.

q Within lexical semanties, especially as applied to computer
Iword meaning is easier when a given word is understood in terms of related

words: semantic networks are therefore important in computational lin

other methods in phonology (e.g. optimality theory, which uses lattice

I morphology (e.g. finite-state morphology, using finite-state transducers) are common
lin the analysis of language as a graph, Indeed, the usefulness of this area of
mathematics to linguistics has bome organizations such as Text Graphs, as well as
[various ‘Net’ projects, such as WorldNet, Verb Net, and others.

Graph theory is also used to study molecules in physics. In condensed
[matter physics, the three-dimensional structure of complicated simulated atomic
structures can be studied quantitatively by gathering statistics on graph-theoretic
[properties related to the topology of the atoms. In statistical physics, graphs can
represent local connections between interacting parts of a system, as well as the

process on such systems. Graphs are also used to represent

the micro-scale channels of porous media, in which the vertices represent the pores
land the edges represent the smaller channels connecting the pores.