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https://gateoverflow.in/questions/mathematics/discrete-mathematics/graph-theory
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https://gateoverflow.in/159638/notes
How many Simple non-isomorphic graphs are possible with vertices 9, edges 9, and degree of each vertex 2?<br />
<br />
an answer is given: 2<br />
<br />
please guide:<br />
<br />
But we can make 3<br />
<br />
1->a cyclic graph of 9 edge<br />
<br />
2-> two cycles one of 5 edge and other of 4 edge<br />
<br />
3-> three cycles of 3 edgeGraph Theoryhttps://gateoverflow.in/159638/notesFri, 13 Oct 2017 07:06:01 +0000Notes
https://gateoverflow.in/159591/notes
How many different (pairwise non-isomorphic trees are there of order 5)?Graph Theoryhttps://gateoverflow.in/159591/notesFri, 13 Oct 2017 04:19:20 +0000GATE2001-2.15 GATE1994-1.6
https://gateoverflow.in/159550/gate2001-2-15-gate1994-1-6
<p>How many undirected graphs are possible with n vertices</p>
<ol style="list-style-type: lower-alpha;">
<li>if graphs are not necessarily connected</li>
<li>if they are necessarily connected</li>
</ol>
<p> </p>Graph Theoryhttps://gateoverflow.in/159550/gate2001-2-15-gate1994-1-6Thu, 12 Oct 2017 17:38:24 +0000schedule graph
https://gateoverflow.in/159384/schedule-graph
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=15245219618812973391"></p>Graph Theoryhttps://gateoverflow.in/159384/schedule-graphThu, 12 Oct 2017 06:28:04 +0000UGCNET-Jan2017-II-5
https://gateoverflow.in/158418/ugcnet-jan2017-ii-5
Consider a Hamiltonian Graph G with no loops or parallel edges and with $|V(G)| = n ≥ 3$. Then which of the following is true ?<br />
<br />
(1) $deg(v) ≥ \frac{n}{2}$ for each vertex v.<br />
<br />
(2) $|E(G)| ≥ \frac{1}{2}(n – 1) (n – 2) + 2$<br />
<br />
(3) $deg (v) + deg(w) ≥ n$ whenever v and w are not connected by an edge.<br />
<br />
(4) All of the aboveGraph Theoryhttps://gateoverflow.in/158418/ugcnet-jan2017-ii-5Sun, 08 Oct 2017 21:11:38 +0000Narsing Deo--Graph theory
https://gateoverflow.in/158248/narsing-deo-graph-theory
<pre>
<code>Show that every planar graph with at least 4 vertices has at least 4
vertices of degree less than or equal to 5.This will also prove there is no 6-connected planar graph.</code></pre>
<p><code>I have proved it partly,can someone please give provide a simple proof to this,especially the 2nd part.</code></p>
<p>Also one doubt,</p>
<p>Sum of degrees of a vertices in graph we know is = 2* no of edges.</p>
<p>Now, Sum of regions of graph (region in terms of graph planarity) is also 2* no of edges.</p>
<p>Is there a interconnection between these terms no just in terms of formula?Can we use it interchangebly.</p>
<p> </p>Graph Theoryhttps://gateoverflow.in/158248/narsing-deo-graph-theorySun, 08 Oct 2017 08:26:43 +0000NIELIT July 2017_74
https://gateoverflow.in/157376/nielit-july-2017_74
<p>If G is an undirected planar graph on <strong>n</strong> vertices with <strong>e</strong> edges then</p>
<p>A) e<=n</p>
<p>B) e<=2n</p>
<p>C) e<=3n</p>
<p>D) None of the option</p>Graph Theoryhttps://gateoverflow.in/157376/nielit-july-2017_74Thu, 05 Oct 2017 04:48:14 +0000NIELIT July 2017_75
https://gateoverflow.in/156941/nielit-july-2017_75
Choose the most appropriate definition of plane graph<br />
<br />
A) A simple graph which is isomorphic to Hamiltonian graph<br />
<br />
B) A graph drawn in a plane such away that if the vertex set of graph can be partitioned into two non - empty disjoint subset X and Y in such a way that each edge of G has a one end in X and one end in Y<br />
<br />
C) A graph drawn in a plane in such a way that any pair of edges meet only at their end vertices<br />
<br />
D) None of the optionGraph Theoryhttps://gateoverflow.in/156941/nielit-july-2017_75Tue, 03 Oct 2017 09:07:13 +0000NIELIT July 2017_72
https://gateoverflow.in/156908/nielit-july-2017_72
<p>The following graph has no Euler circuit because</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=3712102228334608420"></p>
<p>A) It has 7 vertices</p>
<p>B) It is even-valent (all vertices have even valence)</p>
<p>C) It is not connected</p>
<p>D) It does not have an Euler circuit </p>Graph Theoryhttps://gateoverflow.in/156908/nielit-july-2017_72Tue, 03 Oct 2017 06:52:07 +0000NIELIT July 2017_71
https://gateoverflow.in/156905/nielit-july-2017_71
<p>Consider the following graph L and find the bridges, if any</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=17137558656279299350"></p>
<p>A) No bridge</p>
<p>B) {d, e}</p>
<p>C) {c, d} </p>
<p>D) {c, d} and {c, f}</p>Graph Theoryhttps://gateoverflow.in/156905/nielit-july-2017_71Tue, 03 Oct 2017 06:46:15 +0000NIELIT July 2017_73
https://gateoverflow.in/156902/nielit-july-2017_73
<p>For the graph shown, which of teh following paths is a Hamilton circuit ?</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=14213296908841895228"></p>
<p>A) ABCDCFDEFAEA</p>
<p>B) AEDCBAF</p>
<p>C) AEFDCBA</p>
<p>D) AFCDEBA</p>Graph Theoryhttps://gateoverflow.in/156902/nielit-july-2017_73Tue, 03 Oct 2017 06:35:17 +0000Graphs
https://gateoverflow.in/156426/graphs
We know that every 2-colourable graph is bipartite. To prove that we divide the vertices having different colors and put them separately in 2 partitions.<br />
<br />
Suppose there are n vertices in an empty graph and they are randomly colored as 1 and 2. The ones with '1' are placed in partition A and those with '2' are placed in partition B . Can this be called a bipartite case even when there is no connection b/w the vertices of partition A and B? If not, why?Graph Theoryhttps://gateoverflow.in/156426/graphsSat, 30 Sep 2017 19:53:40 +0000UGCNET-june2008-ii-4
https://gateoverflow.in/155561/ugcnet-june2008-ii-4
<p><strong>The set of positive integers under the operation of ordinary multiplication is:</strong></p>
<p><strong>(A) not a monoid (B) not a group</strong></p>
<p><strong>(C) a group (D) an Abelian group</strong></p>Graph Theoryhttps://gateoverflow.in/155561/ugcnet-june2008-ii-4Wed, 27 Sep 2017 08:37:37 +0000Test Series
https://gateoverflow.in/155420/test-series
<p><img alt="" height="403" src="http://gateoverflow.in/?qa=blob&qa_blobid=2453417980509522480" width="718"></p>Graph Theoryhttps://gateoverflow.in/155420/test-seriesTue, 26 Sep 2017 15:25:44 +0000UGCNET-dec2008-ii-2
https://gateoverflow.in/155026/ugcnet-dec2008-ii-2
<p><strong>The graph K<sub>3,4 </sub>has:</strong></p>
<p><strong>(A) 3 edges (B) 4 edges</strong></p>
<p><strong>(C) 7 edges (D) 12 edges</strong></p>Graph Theoryhttps://gateoverflow.in/155026/ugcnet-dec2008-ii-2Mon, 25 Sep 2017 09:40:41 +0000UGCNET-june2009-ii-23
https://gateoverflow.in/154364/ugcnet-june2009-ii-23
<p><strong>Which two of the following are equivalent for an undirected graph G?</strong></p>
<p><strong>(i) G is a tree</strong></p>
<p><strong>(ii) There is at least one path between any two distinct vertices of G</strong></p>
<p><strong>(iii) G contains no cycles and has (n-1) edges</strong></p>
<p><strong>(iv) G has n edges</strong></p>
<p><strong>(A) (i) and (ii)</strong></p>
<p><strong>(B) (i) and (iii)</strong></p>
<p><strong>(C) (i) and (iv)</strong></p>
<p><strong>(D) (ii) and (iii)</strong></p>Graph Theoryhttps://gateoverflow.in/154364/ugcnet-june2009-ii-23Sat, 23 Sep 2017 04:22:59 +0000UGCNET-june2009-ii-12
https://gateoverflow.in/154270/ugcnet-june2009-ii-12
<p><strong>The complete graph with four vertices has k edges where k is:</strong></p>
<p><strong>(A) 3 (B) 4</strong></p>
<p><strong>(C) 5 (D) 6</strong></p>Graph Theoryhttps://gateoverflow.in/154270/ugcnet-june2009-ii-12Fri, 22 Sep 2017 15:32:55 +0000UGCNET-dec2009-ii-22
https://gateoverflow.in/152858/ugcnet-dec2009-ii-22
The number of edges in a complete graph of n vertices is<br />
<br />
(A) n<br />
<br />
(B) n(n – 1)/2<br />
<br />
(C) n(n + 1)/2<br />
<br />
(D) (n^2)/2Graph Theoryhttps://gateoverflow.in/152858/ugcnet-dec2009-ii-22Sun, 17 Sep 2017 07:52:06 +0000[Discrete maths] Spanning trees
https://gateoverflow.in/152732/discrete-maths-spanning-trees
True/False;<br />
<br />
1. Every tree is spanning tree.Graph Theoryhttps://gateoverflow.in/152732/discrete-maths-spanning-treesSat, 16 Sep 2017 16:50:39 +0000UGCNET-dec2009-ii-02
https://gateoverflow.in/152689/ugcnet-dec2009-ii-02
Circle has ____________<br />
<br />
(A) No vertices<br />
<br />
(B) Only 1 vertex<br />
<br />
(C) ∞ vertices<br />
<br />
(D) None of theseGraph Theoryhttps://gateoverflow.in/152689/ugcnet-dec2009-ii-02Sat, 16 Sep 2017 14:27:34 +0000graph - self doubt
https://gateoverflow.in/152154/graph-self-doubt
Graph G has atleast 1 edge then which are true<br />
<br />
a) G has a hamiltonian circuit<br />
<br />
b) Every cycle of G is of even lengthGraph Theoryhttps://gateoverflow.in/152154/graph-self-doubtThu, 14 Sep 2017 12:52:33 +0000graph
https://gateoverflow.in/152152/graph
Assume undirected graph G is connected . G has 6 vertices and 10 edges. Find the minimum number of edges whose deletion from graph G is always guaranteee that it will become disconnectedGraph Theoryhttps://gateoverflow.in/152152/graphThu, 14 Sep 2017 12:48:25 +0000Graph Theory
https://gateoverflow.in/151990/graph-theory
Consider a 'reversed Kruskal' Algorithm for computing a MST. Initialize T to be the set of all edges in the graph. Now consider edges from largest to smallest cost. For each edge, delete it from T if that edge belongs to a cycle in T. Assuming all the edge costs are distinct, does this new algorithm correctly compute a MST?<br />
<br />
<br />
<br />
a) Yes<br />
<br />
b) no<br />
<br />
c) cant sayGraph Theoryhttps://gateoverflow.in/151990/graph-theoryThu, 14 Sep 2017 03:24:49 +0000Graph theory
https://gateoverflow.in/151874/graph-theory
The total number of non isomorphic graph which can be formed with 3 vertices________________________Graph Theoryhttps://gateoverflow.in/151874/graph-theoryWed, 13 Sep 2017 13:19:12 +0000GRAPH THEORY
https://gateoverflow.in/151856/graph-theory
maximum number of edges in a simple planar graph with 10 vertices which contains no triangle is?Graph Theoryhttps://gateoverflow.in/151856/graph-theoryWed, 13 Sep 2017 11:21:56 +0000Graph Theory Question
https://gateoverflow.in/151073/graph-theory-question
<p>Consider a social network with n persons. Two persons A and B are said to be connected if either they are friends or they are related through a sequence of friends: that is, there exists a set of persons F<sub>1</sub>, . . . , Fm such that A and F<sub>1</sub> are friends, F<sub>1</sub> and F<sub>2</sub> are friends, . . . . . . . ., F<sub>m−1</sub> and F<sub>m</sub> are friends, and finally F<sub>m</sub> and B are friends. It is known that there are k persons such that no pair among them is connected. What is the maximum number of friendships possible?</p>Graph Theoryhttps://gateoverflow.in/151073/graph-theory-questionSat, 09 Sep 2017 19:00:47 +0000How many simple path from u to v going through w?
https://gateoverflow.in/150924/how-many-simple-path-from-u-to-v-going-through-w
A complete graph on n vertices is an undirected graph in which every pair of distinct vertices is connected by an edge. A simple path in a graph is one in which no vertex is repeated. Let G be a complete graph on 10 vertices. Let u, v, w be three distinct vertices in G. How many simple paths are there from u to v going through w?<br />
<br />
Please explain in detail. Thank you.Graph Theoryhttps://gateoverflow.in/150924/how-many-simple-path-from-u-to-v-going-through-wSat, 09 Sep 2017 10:25:01 +0000Graph Theory Non isomorphic
https://gateoverflow.in/149457/graph-theory-non-isomorphic
<h1><em><strong>How many number of Non-isomorphism Possible Graph M vertices and E edges ? If any Shortcut Formula </strong></em></h1>Graph Theoryhttps://gateoverflow.in/149457/graph-theory-non-isomorphicSun, 03 Sep 2017 06:54:59 +0000theory of computation
https://gateoverflow.in/146714/theory-of-computation
if L1 ={a^n | n>=0}<br />
<br />
then L1^R is?Graph Theoryhttps://gateoverflow.in/146714/theory-of-computationWed, 23 Aug 2017 13:18:26 +0000Graph theory.
https://gateoverflow.in/145557/graph-theory
Matching and edge coloring are same ?Graph Theoryhttps://gateoverflow.in/145557/graph-theorySat, 19 Aug 2017 15:08:03 +0000SELF DOUBT
https://gateoverflow.in/145488/self-doubt
Explain isomorphism in context of groups with exampleGraph Theoryhttps://gateoverflow.in/145488/self-doubtSat, 19 Aug 2017 10:34:19 +0000Min Cut set
https://gateoverflow.in/145083/min-cut-set
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=731647461265627009"></p>Graph Theoryhttps://gateoverflow.in/145083/min-cut-setFri, 18 Aug 2017 03:45:25 +0000Directed graph
https://gateoverflow.in/144967/directed-graph
how to calculate number of directed graphs possible having n vertices?Graph Theoryhttps://gateoverflow.in/144967/directed-graphThu, 17 Aug 2017 15:03:20 +0000ISI KOLKATA 2017
https://gateoverflow.in/143751/isi-kolkata-2017
<h2>Show that if the edge set of a graph G(V,E) with n nodes can be partitioned into 2 trees then there is at least one vertex of degree less than 4 in G.</h2>Graph Theoryhttps://gateoverflow.in/143751/isi-kolkata-2017Sun, 13 Aug 2017 07:28:41 +0000graph theory
https://gateoverflow.in/143214/graph-theory
For complete bipartite graph k2,3 , what is the edge connectivity ?Graph Theoryhttps://gateoverflow.in/143214/graph-theoryFri, 11 Aug 2017 07:14:47 +0000please solve this Q
https://gateoverflow.in/142533/please-solve-this-q
Q . The maximum number of edges in an undirected graph (simple) with 52 vertices and 3 components areGraph Theoryhttps://gateoverflow.in/142533/please-solve-this-qTue, 08 Aug 2017 18:20:38 +0000Number of Hamiltonian cycles in a complete graph
https://gateoverflow.in/140260/number-of-hamiltonian-cycles-in-a-complete-graph
Number of Hamilton cycles in a complete labelled graph?Graph Theoryhttps://gateoverflow.in/140260/number-of-hamiltonian-cycles-in-a-complete-graphThu, 27 Jul 2017 22:11:35 +0000counting
https://gateoverflow.in/140204/counting
<p>Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to</p>
<ol>
<li>15</li>
<li>30</li>
<li>90</li>
<li>360</li>
</ol>Graph Theoryhttps://gateoverflow.in/140204/countingThu, 27 Jul 2017 11:00:45 +0000planarity is there in syllabus of graph theory or not?
https://gateoverflow.in/138577/planarity-is-there-in-syllabus-of-graph-theory-or-not
Graph Theoryhttps://gateoverflow.in/138577/planarity-is-there-in-syllabus-of-graph-theory-or-notTue, 18 Jul 2017 18:37:22 +0000Please suggest material for graph theory .
https://gateoverflow.in/136778/please-suggest-material-for-graph-theory
Please suggest material for graph theory .Graph Theoryhttps://gateoverflow.in/136778/please-suggest-material-for-graph-theorySat, 08 Jul 2017 17:26:47 +0000Graph Degree sequence : Bondy and Murty : $1.1.16$
https://gateoverflow.in/136078/graph-degree-sequence-bondy-and-murty-%241-1-16%24
Let $d = (d_1,d_2,\dots, d_n)$ be a nonincreasing sequence of nonnegative integers, that is, $d_1 \geq d_2 \geq · · · \geq d_n \geq 0$. Show that:<br />
<br />
there is a loopless graph with degree sequence d if and only if $\sum_{i=1}^{n}d_i$ is even and $d_1 \leq \sum_{i=2}^{n}d_i$Graph Theoryhttps://gateoverflow.in/136078/graph-degree-sequence-bondy-and-murty-%241-1-16%24Wed, 05 Jul 2017 01:06:47 +0000Graph Theory : Bondy-Murty $1.1.20$
https://gateoverflow.in/136075/graph-theory-bondy-murty-%241-1-20%24
<p>Let $S$ be a set of $n$ points in the plane, the distance between any two of which is at least one. Show that there are at most $3n$ pairs of points of S at distance exactly one.</p>
<p> </p>
<p>Can this be done with a <strong>unit circle</strong> and we can place at max. $6$ points on the perimeter and doing the same for other points as well ? i.e. we can get $6n/2 = 3n$ pairs at max. ?
<br>
</p>Graph Theoryhttps://gateoverflow.in/136075/graph-theory-bondy-murty-%241-1-20%24Wed, 05 Jul 2017 00:55:34 +0000Graphic Sequence condition
https://gateoverflow.in/136045/graphic-sequence-condition
<p>A sequence $d = (d_1,d_2,\dots , d_n)$ is <strong><em>graphic</em></strong> if there is a simple graph with degree sequence $d$</p>
<p>If $d = (d_1,d_2,d_3, \dots d_n)$ is graphic and $d_1 \geq d_2 \geq d_3 \geq \dots \geq d_n$ , then show that $\sum_{i=1}^{n}d_i$ is even and $$\sum_{i=1}^{k}d_i \leq \left [ k(k-1) + \sum_{i=k+1}^{n} \min\{k,d_i\} \right ] \quad ,1 \leq k \leq n$$.</p>
<p> </p>Graph Theoryhttps://gateoverflow.in/136045/graphic-sequence-conditionTue, 04 Jul 2017 14:13:13 +0000binary tree - doubt (in solution given in gatecse blog)
https://gateoverflow.in/135989/binary-tree-doubt-in-solution-given-in-gatecse-blog
<p><a rel="nofollow" href="http://gatecse.in/number-of-binary-trees-possible-with-n-nodes/">http://gatecse.in/number-of-binary-trees-possible-with-n-nodes/</a></p>
<p>In the first answer (What is the no. of distinct binary trees possible with n labeled nodes?), </p>
<p>"An edge can be made either as a left child of a node or as a right child. Hence, for n nodes, we have 2n possibilities for the first edge, 2n−1 for the second edge and so on. Thus, for n−1 edges, the total no. of ways..."</p>
<p>I understood above statement. But if we go on and choose like this, what is the surety that we get a tree? </p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=16416592931237639062"></p>
<p>As in the above figure(we choose from 4 nodes in the left side to get the graph in right side), the choices allows us to choose like this,we selected n-1 edges,still we didn't get a tree. </p>Graph Theoryhttps://gateoverflow.in/135989/binary-tree-doubt-in-solution-given-in-gatecse-blogTue, 04 Jul 2017 09:12:05 +0000Minimum No. of vertices required
https://gateoverflow.in/135641/minimum-no-of-vertices-required
<p>Prove the following for graph $G$.</p>
<ol>
<li>When length of the <strong><em>shortest cycle</em></strong> in a graph is $k \geq 3$ and the minimum degree of the graph is $d$, then $G$ has minimum $\begin{align*} \\ 1+ \sum_{0 \leq p < \left \lfloor k/2 \right \rfloor} d\cdot (d-1)^p \end{align*}$ vertices for <strong>odd</strong> $k$.</li>
<li>When the length of the <strong><em>shortest cycle</em></strong> in a graph is $k \geq 4$ and the minimum degree of the graph is $d$, then $G$ has minimum $\begin{align*} \\1+ (d-1)^{\left \lfloor k/2 \right \rfloor -1} + \sum_{0 \leq p < \left \lfloor k/2 \right \rfloor-1} d\cdot (d-1)^p \end{align*}$ vertices for <strong>even </strong>$k$. </li>
</ol>Graph Theoryhttps://gateoverflow.in/135641/minimum-no-of-vertices-requiredSat, 01 Jul 2017 19:27:17 +0000rosen graph theory
https://gateoverflow.in/134449/rosen-graph-theory
find the values (k tuple coloring )<br />
<br />
1)$X_{2}(K_{3}) 2. X_{3}(K_{5})$Graph Theoryhttps://gateoverflow.in/134449/rosen-graph-theoryFri, 23 Jun 2017 07:48:13 +0000Self - Doubt
https://gateoverflow.in/133883/self-doubt
What is clique?Graph Theoryhttps://gateoverflow.in/133883/self-doubtMon, 19 Jun 2017 13:44:27 +0000General Math
https://gateoverflow.in/133745/general-math
Why is $n \leq 2^{h+1} - 1$ equivalent to $h \geq \log_2{\frac{n+1}{2}}$ ? This is applicable to Binary TreesGraph Theoryhttps://gateoverflow.in/133745/general-mathSun, 18 Jun 2017 16:08:26 +0000[Discrete Maths] Graph theory
https://gateoverflow.in/132276/discrete-maths-graph-theory
What is the vertex connectivity and edge connectivity of complete graph?<br />
<br />
Is it n or n-1?Graph Theoryhttps://gateoverflow.in/132276/discrete-maths-graph-theoryWed, 07 Jun 2017 22:18:24 +0000ugc net july 2016
https://gateoverflow.in/131944/ugc-net-july-2016
33. Consider a weighted complete graph G on the vertex set {ν1 , ν2 , …. νn } such that the weight of the edge (νi , νj ) is 4 | i – j|. The weight of minimum cost spanning tree of G is :<br />
<br />
(1) 4n2<br />
<br />
(2) n<br />
<br />
(3) 4n – 4<br />
<br />
(4) 2n – 2Graph Theoryhttps://gateoverflow.in/131944/ugc-net-july-2016Mon, 05 Jun 2017 07:58:13 +0000