This document summarizes applications of graph theory concepts. It discusses using minimum spanning trees for network design problems to connect offices at minimum cost. It also discusses using bipartite graphs and matchings to model job assignment problems, and using graph coloring to create exam schedules where no two exams with a common student occur at the same time. Other applications mentioned include using planar graphs in circuit board design and transportation planning.

1. APPLIED GRAPH THEORY
ASSIGNMENT NO-2
SUBMITTED TO
DR.SURENDRA SINGH
SUBMITTED BY
SANESH KUMAR
PRIYADARSHAN KUMAR
SHUBHAM TIWARI
AMAN TIWARI
2011ECS25
2011ECS35
(2011ECS27)
(2011ECS55)

2. 1
Graph covering in electrical network
Loop Currents and Node Voltages
Consider the vector space W. (over the field of real numbers) associated
with the directed graph G. Here G is a connected directed graph of e edges
and n vertices, representing an electrical network. From Eq. (13-2), we see
that the edge-current vector ((t) is orthogonal to each of the row vectors in
the incidence matrix A. Since the row vectors in A span the entire cut-set
subspace Ws (of dimension n — 1), ((t) is orthogonal to W. Therefore, ((t)
lies in the circuit subspace IV, (of dimension /1 =e —s -) I) of G.
Since i(t) is contained in Wr, there must be a set of p vectors in Wr whose
linear combination will produce i(t). An obvious choice for this set of p
linearly independent vectors in Wr is the rows of the fundamental circuit
matrix Bf with respect to some spanning tree. (Clearly, Bf is contained in
B.) Let the coordinates (or coefficients) of i(t) in this basis formed by the
rows b,, b, b, of 13, be 4,0), i„(t) ..... 1„(t). In other words,
Thus each of the e edge currents can be expressed as a linear combination
of g quantities Ur), 44(0, . . . , i(t). The are called hoop currents (or mesh
currents); they represent current flowing in the p independent circuits
corresponding to the rows of B,. Substituting Eq. (13-5) into Eq. (13-2),
we get
(AB))iL(t) = 0. (13-
6)
Similarly, from Eq. (13-4) we see that the column vector v(t) representing
the edge voltage is orthogonal to the circuit subspace Weed is, therefore,
in cut-set subspace W,. Thus v(t) can be expressed as a linear combination
of the n — 1 rows of the reduced incidence matrix A. That is,

3. 2

4. 3

5. 4
Network flow
Minimum Spanning Tree
Minimum Spanning Tree (MST) problem: Given connected graph G with
positive edge
weights, find
a min weight
set of edges
that connects
all of the
vertices.
Network design
The standard application is to a problem like phone network design. We
have a business with several offices; we want to lease phone lines to
connect them up with each other; and the phone company charges

6. 5
different amounts of money to connect different pairs of cities. We want a
set of lines that connects all our offices with a minimum total cost. It
should be a spanning tree, since if a network isn’t a tree, we can always
remove some edges and save money.
Bipartite graph in job assignment problem
Task: Given a weighted complete bipartite graph G = (X ׫Y,X ×Y ),
where edge xy has weight w(xy), find a matching M from X to Y with
maximum weight.
In an application, X could be a set of workers, Y could be a set of jobs, and
w(xy) could be the profit made by assigning worker x to job y.
By adding virtual jobs or workers with 0 profitability, we may assume that
X and Y have the same size, n, and can be written as X = {x1,x2,...,xn}
and Y = {y1,y2,...,yn}. Mathematically, the problem can be stated: given
an n×n matrix W, find a permutation ʌ of {1,2,3,...,n} for which n X i=1
w(xiyʌ(i)) is a maximum.
Such a matching from X to Y is called an optimal assignment.
Definition: A feasible vertex labeling in G is a real-valued function l on X
׫Y such that for all x א X and y א Y , l(x) + l(y) ≥ w(xy). Such labelings
may be conveniently written beside the matrix of weights.
It is always possible to find a feasible vertex labeling. One way to do this
is to set all l(y) = 0 for y א Y and for each x א X, take the maximum value
in the corresponding row, i.e. l(x) = maxyאY w(xy) for x א X l(y) = 0 for
y א Y
If l is a feasible labeling, we denote by Gl the subgraph of G which
contains those edges where l(x) + l(y) = w(xy), together with the endpoints
of these edges. This graph Gl is called the equality subgraph for l.
Theorem. If l is a feasible vertex labeling for G, and M is a complete
matching of X to Y with M ك Gl, then M is an optimal assignment of X to
Y .
Proof: We must show that no other complete matching can have weight
greater than M. Let any complete matching M0 of X to Y be given. Then
w(M0) = X xyאM0 w(xy) ≤ X xyאM0 (l(x) + l(y)) (feasibility of l) = X
xyאM (l(x) + l(y)) (all the l(x) and l(y) are summed in either matching) =
X xyאM w(xy) since M ك Gl = w(M).

7. 6
Thus the problem of finding an optimal assignment is reduced to the
problem of finding a feasible vertex labeling whose equality subgraph
contains a complete matching of X to Y.
Bipartite graph in Matching problem
Maximum Bipartite Matching
A matching in a Bipartite Graph is a set of the edges chosen in such a way
that no two edges share an endpoint. A maximum matching is a matching
of maximum size (maximum number of edges). In other words, a
matching is maximum if any edge is added to it, it is no longer a matching.
There can be more than one maximum matching for a given Bipartite
Graph.
Problem: There are M job applicants and N jobs. Each applicant has a subset
of jobs that he/she is interested in. Each job opening can only accept one
applicant and a job applicant can be appointed for only one job. Find an
assignment of jobs to applicants in such that as many applicants as possible get
jobs.
Solution: This can be solved by Ford-Fulkerson algorithm.

8. 7
Application of planar graph
• Easy to visualize. In fact, crossing of edges is the main culprit for
reducing comprehensibility.
• VLSI design, circuit needs to be on surface: lesser the crossings,
better is the design.
• High-speed Highways/Railroads design, crossings are always
problematic.
• Irrigation canals, crossings simply not admissible.
Time table schedule
An application of Graph Coloring.
Making Schedule or Time Table: Suppose we want to make am exam
schedule for a university. We have list different subjects and students
enrolled in every subject. Many subjects would have common students (of
same batch, some backlog students, etc). How do we schedule the exam so
that no two exams with a common student are scheduled at same time?
How many minimum time slots are needed to schedule all exams? This
problem can be represented as a graph where every vertex is a subject and
an edge between two vertices mean there is a common student. So this is a
graph coloring problem where minimum number of time slots is equal to
the chromatic number of the graph.
Bio- informatics
Famous Problem: Shortest Path
• Greedy strategy works for simple problems (no cycles, no negative
weights)
• Longest path is a similar problem (complement
weights)
Solution:

9. 8
DIJKASTRA’S ALGORITHM
References
• http://www.academia.edu/1325212/Timetable_Schedulin
g_using_Graph_Coloring
• http://www.math.uwo.ca/~mdawes/courses/344/kuhnmunk
res.pdf
• http://www.tcs.tifr.res.in/~workshop/nitp_igga/slides/shr
eesh-planarity-patna.pdf
• http://www.geeksforgeeks.org/graph-coloringapplications/
• http://www.geeksforgeeks.org/maximum-bipartitematching/
• http://www.geeksforgeeks.org/ford-fulkerson-algorithmfor-
maximum-flow-problem/
• http://www.slideshare.net/UmangGupta5/graph-theorynarsingh-
deo

10. 9
• https://www.google.co.in/url?sa=t&rct=j&q=&esrc=s&s
ource=web&cd=4&ved=0CDkQFjAD&url=http%3A%2
F%2Fwww.geeksforgeeks.org%2Fmaximum-bipartitematching%
2F&ei=mSFfVODlI8fVuQTMloDYC
Q&usg
=AFQjCNFC68CkA7s0oszG5T9frWHVah1xEg&sig2=
Q18G5RJPuMNWwCbXwoziRQ&bvm=bv.79189006,d.
c2E&cad=rja