GATE Overflow - Recent questions in Graph Theory
http://gateoverflow.in/questions/mathematics/discrete-mathematics/graph-theory
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http://gateoverflow.in/145083/min-cut-set
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=731647461265627009"></p>Graph Theoryhttp://gateoverflow.in/145083/min-cut-setFri, 18 Aug 2017 03:45:25 +0000Directed graph
http://gateoverflow.in/144967/directed-graph
how to calculate number of directed graphs possible having n vertices?Graph Theoryhttp://gateoverflow.in/144967/directed-graphThu, 17 Aug 2017 15:03:20 +0000ISI KOLKATA 2017
http://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 Theoryhttp://gateoverflow.in/143751/isi-kolkata-2017Sun, 13 Aug 2017 07:28:41 +0000graph theory
http://gateoverflow.in/143214/graph-theory
For complete bipartite graph k2,3 , what is the edge connectivity ?Graph Theoryhttp://gateoverflow.in/143214/graph-theoryFri, 11 Aug 2017 07:14:47 +0000please solve this Q
http://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 Theoryhttp://gateoverflow.in/142533/please-solve-this-qTue, 08 Aug 2017 18:20:38 +0000Number of Hamiltonian cycles in a complete graph
http://gateoverflow.in/140260/number-of-hamiltonian-cycles-in-a-complete-graph
Number of Hamilton cycles in a complete labelled graph?Graph Theoryhttp://gateoverflow.in/140260/number-of-hamiltonian-cycles-in-a-complete-graphThu, 27 Jul 2017 22:11:35 +0000counting
http://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 Theoryhttp://gateoverflow.in/140204/countingThu, 27 Jul 2017 11:00:45 +0000planarity is there in syllabus of graph theory or not?
http://gateoverflow.in/138577/planarity-is-there-in-syllabus-of-graph-theory-or-not
Graph Theoryhttp://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 .
http://gateoverflow.in/136778/please-suggest-material-for-graph-theory
Please suggest material for graph theory .Graph Theoryhttp://gateoverflow.in/136778/please-suggest-material-for-graph-theorySat, 08 Jul 2017 17:26:47 +0000Graph Degree sequence : Bondy and Murty : $1.1.16$
http://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 Theoryhttp://gateoverflow.in/136078/graph-degree-sequence-bondy-and-murty-%241-1-16%24Wed, 05 Jul 2017 01:06:47 +0000Graph Theory : Bondy-Murty $1.1.20$
http://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 Theoryhttp://gateoverflow.in/136075/graph-theory-bondy-murty-%241-1-20%24Wed, 05 Jul 2017 00:55:34 +0000Graphic Sequence condition
http://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 Theoryhttp://gateoverflow.in/136045/graphic-sequence-conditionTue, 04 Jul 2017 14:13:13 +0000binary tree - doubt (in solution given in gatecse blog)
http://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 Theoryhttp://gateoverflow.in/135989/binary-tree-doubt-in-solution-given-in-gatecse-blogTue, 04 Jul 2017 09:12:05 +0000Minimum No. of vertices required
http://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 Theoryhttp://gateoverflow.in/135641/minimum-no-of-vertices-requiredSat, 01 Jul 2017 19:27:17 +0000rosen graph theory
http://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 Theoryhttp://gateoverflow.in/134449/rosen-graph-theoryFri, 23 Jun 2017 07:48:13 +0000Self - Doubt
http://gateoverflow.in/133883/self-doubt
What is clique?Graph Theoryhttp://gateoverflow.in/133883/self-doubtMon, 19 Jun 2017 13:44:27 +0000General Math
http://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 Theoryhttp://gateoverflow.in/133745/general-mathSun, 18 Jun 2017 16:08:26 +0000[Discrete Maths] Graph theory
http://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 Theoryhttp://gateoverflow.in/132276/discrete-maths-graph-theoryWed, 07 Jun 2017 22:18:24 +0000ugc net july 2016
http://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 Theoryhttp://gateoverflow.in/131944/ugc-net-july-2016Mon, 05 Jun 2017 07:58:13 +0000graphtheory,Narsingh Deo,4.26
http://gateoverflow.in/130286/graphtheory-narsingh-deo-4-26
Suppose a single tennis tournament is arranged among n players and the number of matches planned is a fixed number e (where n-1 < e < n(n-1)/2 ).For sake of fairness,how will you make sure that some players do not group together and isolate an individual (or a group of players).Graph Theoryhttp://gateoverflow.in/130286/graphtheory-narsingh-deo-4-26Sun, 21 May 2017 03:34:20 +0000#Graphtheory
http://gateoverflow.in/130285/%23graphtheory
Construct a graph G with edge connectivity of G =4 ,vertex connectivity of G =3 and degree of every vertex of G >=5Graph Theoryhttp://gateoverflow.in/130285/%23graphtheorySun, 21 May 2017 02:56:10 +00002 - connected graph
http://gateoverflow.in/130141/2-connected-graph
<p>For a <strong>regular graph</strong> how much large the value of degree (for each vertices) should be such that the graph is $2$ - connected. (vertex wise).</p>
<p>I did in this way :</p>
<p>$\begin{align*} &\quad \kappa(G) \leq \frac{2\cdot e}{n} \qquad \text{ where } \kappa(G) = \text{ vertex connectivity } \\ &\Rightarrow 2 \leq \frac{2\cdot e}{n} \\ &\Rightarrow n \leq e \\ &\Rightarrow n \leq \frac{\sum \left ( d_i \right )}{2} \\ &\Rightarrow n \leq \frac{n \cdot d}{2} \\ &\Rightarrow d \geq 2 \\ \end{align*}$</p>
<p>The above case can be realized by thinking of a <strong>cycle graph</strong> of $n$ vertices.</p>
<p>But in the following case :</p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=15360590681287688332"></p>
<p>This graph is 3 regular and not 2- connected although $d \geq 2$ is satisfied.</p>
<p>Why this $d \geq 2$ is trivial and not working in some cases ?</p>Graph Theoryhttp://gateoverflow.in/130141/2-connected-graphFri, 19 May 2017 07:12:05 +0000Isomorphism and subgraph
http://gateoverflow.in/130078/isomorphism-and-subgraph
If there are two graphs G1 and G2 and both are Isomorphic to each other...Is G1 subset of G2?Graph Theoryhttp://gateoverflow.in/130078/isomorphism-and-subgraphThu, 18 May 2017 15:21:51 +0000find complexity?
http://gateoverflow.in/129573/find-complexity
What is cyclomatic complexity of below program below:<br />
<br />
i=0; n=4;<br />
<br />
while(i<n-1)<br />
<br />
{j=i+1;<br />
<br />
while(j<n)<br />
<br />
{if(A[i]< A[j]) swap (A[i],A[j]);}<br />
<br />
i=i+1;<br />
<br />
}<br />
<br />
A)3 B)4 C)5 D)6Graph Theoryhttp://gateoverflow.in/129573/find-complexitySun, 14 May 2017 04:03:04 +0000GATE Graph Theory
http://gateoverflow.in/129447/gate-graph-theory
Let G = (V, E) be a directed graph where V is the set of vertices and E the set of edges. Then which one of the following graphs has the same strongly connected components as G?<br />
<br />
( A ) G1 = (V, E1) where E1 = {(u, v) | (u, v) ∉ E}<br />
( B ) G2 = (V, E2) where E2 = {(u, v) | (v, u) ∉ E}<br />
( C ) G3 = (V, E3) where E3 = {(u, v) | there ish a path of length ≤ 2 from u to v in E}<br />
( D ) G4 = (V4, E) where V4 is the set of vertices in G which are not isolated<br />
<br />
Can anyone give a detailed answer to this question, please? :)Graph Theoryhttp://gateoverflow.in/129447/gate-graph-theoryFri, 12 May 2017 19:24:26 +0000Graph Theory
http://gateoverflow.in/129355/graph-theory
algorithm to find more than one path between any two vertices of a graph G=(V,E) , with a complexity of O(VE) ?Graph Theoryhttp://gateoverflow.in/129355/graph-theoryFri, 12 May 2017 08:04:59 +0000Graph theory
http://gateoverflow.in/129113/graph-theory
Which is the best book for studying GATE CS Graph theory?Graph Theoryhttp://gateoverflow.in/129113/graph-theoryWed, 10 May 2017 04:00:03 +0000Graph theory and Applications Bondy and Murty Exercise Qn 1.9
http://gateoverflow.in/129036/graph-theory-and-applications-bondy-and-murty-exercise-qn-1
<p>A k partite graph is one where vertex set can be partitioned into k subsets so that no edge has both end in any one subset.</p>
<p>A complete k partite graph is one that is simple and in which each vertex is joined to every other vertex that is not in the same subset. The complete m-partite graph on n vertices in which each part has either floor(n/m) or ceil(n/m) vertices is denoted by T<sub>m,n</sub> . Show that</p>
<p>a) | E(T<sub>m,n</sub>) | = $\binom{n-k}{2} + (m-1)\binom{k+1}{2} , k = \left \lfloor n/m \right \rfloor$</p>
<p>b) If G is a complete m-partite graph on n vertices then | E(G) | <= | E(T<sub>m,n</sub>)|, with equality only if G isomorphic to T<sub>m,n</sub></p>Graph Theoryhttp://gateoverflow.in/129036/graph-theory-and-applications-bondy-and-murty-exercise-qn-1Tue, 09 May 2017 16:09:49 +0000keneth r rosen
http://gateoverflow.in/128471/keneth-r-rosen
how to prove that sum of all the vertices in a graph G is equal to twice the number of edges in G.<br />
<br />
please explain step by step .Graph Theoryhttp://gateoverflow.in/128471/keneth-r-rosenSun, 07 May 2017 13:28:40 +0000keneth r rosen
http://gateoverflow.in/128470/keneth-r-rosen
<p>how to prove that graph G with<strong> </strong>e= v - 1 that has no circuit is a tree.</p>Graph Theoryhttp://gateoverflow.in/128470/keneth-r-rosenSun, 07 May 2017 13:26:36 +0000PGEE 2017
http://gateoverflow.in/127521/pgee-2017
Consider a graph where vertex having number 2 to 12 (including 2 and 12), there is an edge between two vertex x and y iff x divides y<br />
<br />
What would be maximum path length between any two vertices of graph ?Graph Theoryhttp://gateoverflow.in/127521/pgee-2017Sun, 30 Apr 2017 18:04:30 +0000PGEE 2017
http://gateoverflow.in/127520/pgee-2017
Consider a graph where vertex having number 2 to 12 (including 2 and 12), there is an edge between two vertex x and y iff x divides y<br />
<br />
Which vertex will have highest in degree ?Graph Theoryhttp://gateoverflow.in/127520/pgee-2017Sun, 30 Apr 2017 18:00:23 +0000PGEE 2017
http://gateoverflow.in/127519/pgee-2017
Consider a graph where vertex having number 2 to 12 (including 2 and 12), there is an edge between two vertex x and y iff x divides y<br />
<br />
Find number of strongly connected componentsGraph Theoryhttp://gateoverflow.in/127519/pgee-2017Sun, 30 Apr 2017 17:58:18 +0000Nonisomorphic Graphs
http://gateoverflow.in/127496/nonisomorphic-graphs
How many nonisomorphic connected simple graphs are<br />
there with 5 vertices?Graph Theoryhttp://gateoverflow.in/127496/nonisomorphic-graphsSun, 30 Apr 2017 16:15:23 +0000Paths in a graph
http://gateoverflow.in/127187/paths-in-a-graph
The number of paths of length 5 between two different<br />
vertices in K4 (complete graph)?Graph Theoryhttp://gateoverflow.in/127187/paths-in-a-graphFri, 28 Apr 2017 06:54:07 +0000Rosen discrete mathematics
http://gateoverflow.in/127186/rosen-discrete-mathematics
a)The number of paths of length 4 between any two<br />
adjacent vertices in K3,3 (bipartite graph)?<br />
<br />
b) The number of paths of length 4 between any two<br />
nonadjacent vertices in K3,3 (bipartite graph)?Graph Theoryhttp://gateoverflow.in/127186/rosen-discrete-mathematicsFri, 28 Apr 2017 06:53:43 +0000Self-Doubt
http://gateoverflow.in/126868/self-doubt
Every Planar graph have vertex cover of size atmost 3n/4.<br />
<br />
Can someone provide a good link to understand the above fact?<br />
<br />
Or a good explanation is most welcome.Graph Theoryhttp://gateoverflow.in/126868/self-doubtTue, 25 Apr 2017 06:41:09 +0000graph theory
http://gateoverflow.in/124669/graph-theory
A graph consists of only one vertex,which is isolated ..Is that graph<br />
<br />
A) a complete graph ???<br />
<br />
B) a clique???<br />
<br />
C) connected graph ???<br />
<br />
Please explain your answer ...Graph Theoryhttp://gateoverflow.in/124669/graph-theoryFri, 07 Apr 2017 17:38:26 +0000ISI Entrance Exam MTech (CS)
http://gateoverflow.in/124367/isi-entrance-exam-mtech-cs
Consider all possible trees with $n$ nodes. Let $k$ be the number of nodes with degree greater than $1$ in a given tree. What is the maximum possible value of $k$?Graph Theoryhttp://gateoverflow.in/124367/isi-entrance-exam-mtech-csThu, 06 Apr 2017 00:52:41 +0000JNUEE-2016
http://gateoverflow.in/123807/jnuee-2016
Consider an undirected graph G with 100 nodes. What is the maximum number of edges to be included in G so that graph is connected?<br />
<br />
(a) 2451<br />
<br />
(b) 4851<br />
<br />
(c) 4950<br />
<br />
(d) 9990Graph Theoryhttp://gateoverflow.in/123807/jnuee-2016Mon, 03 Apr 2017 21:09:51 +0000cil 2017
http://gateoverflow.in/123566/cil-2017
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=12137641003618320025"></p>
<p>a. 2</p>
<p>b. 4</p>
<p>c. 1</p>
<p>d. 3</p>Graph Theoryhttp://gateoverflow.in/123566/cil-2017Sun, 02 Apr 2017 17:28:21 +0000Graph Theory Path
http://gateoverflow.in/123440/graph-theory-path
What is the difference between path and Euler path?Graph Theoryhttp://gateoverflow.in/123440/graph-theory-pathSun, 02 Apr 2017 02:20:33 +0000Consider a graph of only 1 vertex and no edges. Is it connected or disconnected?
http://gateoverflow.in/122976/consider-graph-only-vertex-and-edges-connected-disconnected
Graph Theoryhttp://gateoverflow.in/122976/consider-graph-only-vertex-and-edges-connected-disconnectedThu, 30 Mar 2017 13:23:42 +0000Suppose a graph G with n vertices is isomorphic to its complement.How many edges does G have?
http://gateoverflow.in/122925/suppose-graph-vertices-isomorphic-complement-many-edges-does
Graph Theoryhttp://gateoverflow.in/122925/suppose-graph-vertices-isomorphic-complement-many-edges-doesWed, 29 Mar 2017 23:01:21 +0000Graph_Theory
http://gateoverflow.in/122276/graph_theory
Determine all non- isomorphic graphs with the number of vertices 20 and edges 188.Graph Theoryhttp://gateoverflow.in/122276/graph_theoryTue, 21 Mar 2017 15:15:08 +0000No of spanning Trees
http://gateoverflow.in/122066/no-of-spanning-trees
Let $K_n$ denote the complete undirected graph with $n$ vertices where n is an even number. Find the maximum number of spanning trees of $K_n$ that can be formed in such a way that no two of these spanning trees have a common edge.Graph Theoryhttp://gateoverflow.in/122066/no-of-spanning-treesSun, 19 Mar 2017 13:10:09 +0000radius ,diameter of graph
http://gateoverflow.in/121818/radius-diameter-of-graph
The distance between two distinct vertices v1 and v2 of a<br />
connected simple graph is the length (number of edges) of<br />
the shortest path between v1 and v2. The radius of a graph<br />
is the minimum over all vertices v of the maximum distance<br />
from v to another vertex. The diameter of a graph<br />
is the maximum distance between two distinct vertices.<br />
Find the radius and diameter of<br />
a) K6. b) K4,5. c) Q3. d) C6.Graph Theoryhttp://gateoverflow.in/121818/radius-diameter-of-graphThu, 16 Mar 2017 16:53:23 +0000Rosen ex.55 chp 8
http://gateoverflow.in/121595/rosen-ex-55-chp-8
If the simple graph G has v vertices and e edges, how many edges does G complement have?Graph Theoryhttp://gateoverflow.in/121595/rosen-ex-55-chp-8Tue, 14 Mar 2017 21:41:53 +0000graph theory
http://gateoverflow.in/121379/graph-theory
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=14124213158983667542"></p>
<p><img alt="" src="http://gateoverflow.in/?qa=blob&qa_blobid=5624474628795835148"></p>Graph Theoryhttp://gateoverflow.in/121379/graph-theorySun, 12 Mar 2017 17:52:08 +0000graph theory
http://gateoverflow.in/121304/graph-theory
<p><strong>chromatic number of a graph <= ( maxdegree of the graph ) + 1 </strong></p>
<p>can somebody explain how ?</p>Graph Theoryhttp://gateoverflow.in/121304/graph-theorySat, 11 Mar 2017 16:53:30 +0000