C.U.SHAH UNIVERSITY Winter Examination-2015 Subject Name : Graph Theory Subject Code : 5SC03GTE1 Semester : III
Branch : M.Sc (Mathematics)
Date: 08/12/2015
Time : 2:30 To 5:30
Marks : 70_
Instructions: (1) Use of Programmable calculator and any other electronic instrument is prohibited. (2) Instructions written on main answer book are strictly to be obeyed. (3) Draw neat diagrams and figures (if necessary) at right places. (4) Assume suitable data if needed.
SECTION – I Q-1
Attempt the Following questions.
a. Give an example of graph which has a Hamiltonian circuit but not an Euler circuit. b. Draw the undirected graph represented by incidence matrix given below. 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 c. How many edges are there in undirected graph with 6 vertices each of degree 5? d. Define Euler graph. Q-2 Attempt all questions From the digraph given below, answer the following: A 1. Find in degree, out degree and total degree of each vertex. 2. Find reachable set of each vertex. 3. Find all node bases. 4. Find all strong components. 5. Write the adjacency matrix.
B
Prove that all paths are elementary in a tree. Page 1 || 4
(02) (02)
(02) (01) (07)
(07)
Q-2 A
OR Attempt all questions Give three different representations of a tree from
(07)
( v ( v ( v ) ( v ( v )( v ))) ( v ( v ( v )) ( v )( v ))) 0
1
2
3
4
5
6
7
8
9
10
Also identify root, branch nodes and leaf nodes from the tree. B Q-3 A
B
Q-3 A
B
Prove that a given connected graph G is an Euler graph if and only if all vertices of G are of even degree. Attempt all questions Obtain binary tree equivalent to the tree given below:
Prove that an arborescence is a tree in which every vertex other then the root has an in degree of exactly one. OR Attempt all questions Define Spanning tree and draw all possible spanning trees of the graph given below:
Prove that in a complete graph with n vertices there are circuits if n is an odd number ≥ 3.
Page 2 || 4
edge – disjoint Hamiltonian
(07)
(07)
(07)
(07)
(07)
SECTION – II Q-4 a.
Attempt the Following questions. Define proper coloring and determine chromatic number of the graph given below.
b. State Hall’s matching condition. c. For the bipartite graph given below, find an independence number.
d. Q-5 A B
Q-5 B
B
Define: Edge cover. Attempt all questions Prove that every tree with two or more vertices is 2 – chromatic. Define minimum spanning tree and determine minimum spanning tree for the graph given below.
(02)
(02) (02)
(01) (07) (07)
OR Attempt all questions Define chromatic polynomial and find the chromatic polynomial of the graph given below:
(07)
Prove that for every k – regular bipartite graph has a perfect matching. Where k > 0.
(07)
Page 3 || 4
Q-6 A
B
Attempt all questions Define adjacency matrix of a digraph and obtain adjacency matrix for the graph given below:
(07)
Prove that an n-vertex graph is a tree if and only if its chromatic polynomial
(07)
= Q-6 A
B
− 1)
OR Attempt all Questions For a bipartite graph given below, Answer the following: (i) Find complete matching of set V1 into V2 . (ii) Find deficiency. (iii) Determine maximum number of vertices in set V1 that can be matched into V2 . (iv) Find adjacency matrix of a graph. (v) Represent complete matching of V1 into V2 in matrix form.
If G is a bipartite graph then prove that the maximum size of a matching in G equals the minimum size of a vertex cover of G.
Define proper coloring and determine chromatic number of the graph given below. (02). b. State Hall's matching condition. (02). c. For the bipartite graph given ...
The web is a vast repository of knowledge, but automatically extracting that ... Early work on the problem of jointly identifying a best latent KB from a collec- ... limitations, and we build on and improve the model of Jiang et al. by including ....
(c) Find the chromatic number of the Graph 4. given below. If the chromatic number is k, ... MMTE-001 3. Page 3 of 5. Main menu. Displaying Graph Theory.PDF.
ample, in World Wide Web, dense subgraphs might be communities or link spam; in telephone call graph, dense subgraphs might be groups of friends or families. In these situations, the graphs are usually very sparse in global, but have many dense subgr
To prove the minimality of the set MFIS(X), we will show that for any set N ..... Prove that for a non-empty regular bipartite graph the number of vertices in both.
Let us define A = {v1,...,vm} and B = V (G)âA. We split the sum m. â i=1 di into two parts m. â i=1 di = C + D, where C is the contribution of the edges with both ...
Models for small world? ▫ Erdos-Renyi model. ▫ n nodes, each node has a probability p of ... Barabasi-Albert model. ▫ Graph not static, but grows with time.
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Read Algorithms in C++ Part 5: Graph Algorithms: Graph Algorithms Pt.5 - Online. Book detail. Title : Read Algorithms in C++ Part 5: Graph q. Algorithms: Graph ...
IBM T. J. Watson Research Center. â¡ ... gantic databases. ..... We call the code learning model formulated in Eq. (4) as Discrete Graph Hashing (DGH). Because.
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Abstract. Graphs have a wide range of applications in many domains. The graph substructure selection problem is to find all subgraph isomor- phic mappings of ...
tention recently, in the context of analyzing social networks and the World ..... [10]. We consider the converse of the densest k-subgraph problem, in which the.
AbstractâWe introduce the notion of information ratio. Ir(H/G) between two (simple, undirected) graphs G and H, which characterizes the maximal number of source symbols per channel use that can be reliably sent over a channel with confusion graph H
There was a problem previewing this document. Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. Practice Graph ...
The degree of v â V (G), denoted deg(v), is the number of edges incident with v. Alterna- tively, deg(v) = |N(v)|. Definition 3 The complement of a graph G = (V,E) is a graph with vertex set V and edge set. E such that e â E if and only if e â
TrigCheatSheet.com. Page 1 of 1. Graph Paper Full.pdf. Graph Paper Full.pdf. Open. Extract. Open with. Sign In. Main menu. Displaying Graph Paper Full.pdf.
mial time. Since then, many problems have been shown to reduce to graph cut problems ...... Technical Report 2002-06, NEC, Princeton, 2002. 16. L. Ford and ...
matrix trace norm, matrix Frobenius norm, l1 norm, and inner-product operator, respectively. Anchor Graphs. In the discrete graph hashing model, we need to ...
Until we define the notion of weak closedness, we fix a graph G and a labeling of. V (G). Let (a1,...,an) be a sequence such that 1 ⤠ai ⤠n and ai = aj if i = j. Definition 2.1. We say that ai is interchangeable with ai+1 if {ai,ai+1} â E(G).
