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GATE CSE
Graph Theory
Discrete Mathematics
Previous Years Questions
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Marks 1
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Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?...
GATE CSE 2022
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The following simple undirected graph is referred to as the Peterson graph. Which of the following statements is/are T...
GATE CSE 2022
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Consider a simple undirected weighted graph G, all of whose edge weights are distinct. Which of the following statements...
GATE CSE 2022
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Consider a simple undirected unweighted graph with at least three vertices. If A is the adjacency matrix of the graph, t...
GATE CSE 2022
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Consider a simple undirected graph of 10 vertices. If the graph is disconnected, then the maximum number of edges it can...
GATE CSE 2022
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Which of the following statements is/are TRUE for a group G?
GATE CSE 2022
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Let G be a connected undirected weighted graph. Consider the following two statements. S1: There exists a minimum weigh...
GATE CSE 2021 Set 2
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Let G be an undirected complete graph on n vertices, where n > 2. Then, the number of different Hamiltonian cycles in...
GATE CSE 2019
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The chromatic number of the following graph is _______. ...
GATE CSE 2018
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Let $$G$$ be a finite group on $$84$$ elements. The size of a largest possible proper subgroup of $$G$$ is ________.
GATE CSE 2018
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The minimum number of colours that is sufficient to vertex-colour any planar graph is _____________ .
GATE CSE 2016 Set 2
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let $$G$$ be a group with $$15$$ elements. Let $$L$$ be a subgroup of $$G$$. It is known that $$L \ne G$$ and that the s...
GATE CSE 2014 Set 3
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The maximum number of edges in a bipartite graph on $$12$$ vertices is _________.
GATE CSE 2014 Set 1
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Let $$G = \left( {V,E} \right)$$ be a directed graph where $$V$$ is the set of vertices and $$E$$ the set of edges. Then...
GATE CSE 2014 Set 1
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Which of the following statements is/are TRUE for undirected graphs? P: Number of odd degree vertices is even. Q: ...
GATE CSE 2013
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Consider an undirected random$$^ \circ $$ graph of eight vertices. The probability that there is an edge between a pair ...
GATE CSE 2013
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Let G be a simple undirected planner graph on 10 vertices with 15 edges. If G is a connected graph, then the number of b...
GATE CSE 2012
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$$K4$$ and $$Q3$$ are graphs with the following structures. Which one of the following statements is TRUE in relation t...
GATE CSE 2011
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Let $$G$$ $$\,\,\,\,\, = \,\,\,\left( {V,\,\,\,\,\,E} \right)$$ be a graph. Define $$\xi \left( G \right) = \sum\limits_...
GATE CSE 2010
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Which one of the following is TRUE for any simple connected undirected graph with more than $$2$$ vertices?
GATE CSE 2009
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What is the chromatic number of an $$n$$-vertex simple connected graph which does not contain any odd length cycle? Assu...
GATE CSE 2009
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What is the chromatic number of the following graph? ...
GATE CSE 2008
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What is the size of the smallest MIS (Maximal Independent Set) of a chain of nine nodes?
GATE CSE 2008
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The maximum number of binary trees that can be formed with three unlabeled nodes is:
GATE CSE 2007
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The height of a binary tree is the maximum number of edges in any root to leaf path. The maximum number of nodes in a bi...
GATE CSE 2007
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Consider a weighted undirected graph with positive edge weights and let $$uv$$ be an edge in the graph. It is known that...
GATE CSE 2007
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Let $$G$$ be the non-planar graph with minimum possible number of edges. Then $$G$$ has
GATE CSE 2007
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Consider a weighted complete graph $$G$$ on the vertex set $$\left\{ {{v_1},\,\,\,{v_2},....,\,\,\,{v_n}} \right\}$$ suc...
GATE CSE 2006
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If all the edge weights of an undirected graph are positive, then any subject of edges that connects all the vertices an...
GATE CSE 2006
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Let $$G$$ be the simple graph with 20 vertices and 100 edges. The size of the minimum vertex cover of $$G$$ is 8. Then, ...
GATE CSE 2005
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Let $$G$$ be a simple connected planar graph with 13 vertices and 19 edges. Then, the number of faces in the planar embe...
GATE CSE 2005
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What is the maximum number of edges in an acyclic undirected graph with $$n$$ vertices?
GATE CSE 2004
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Let $$G$$ be an arbitrary graph with $$n$$ nodes and $$k$$ components. If a vertex is removed from $$G$$, the number of ...
GATE CSE 2003
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Maximum number of edges in a n - node undirected graph without self loops is
GATE CSE 2002
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The number of distinct simple graph with upto three nodes is
GATE CSE 1994
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Which of the following is/are tautology?
GATE CSE 1992
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A non-planar graph with minimum number of vertices has
GATE CSE 1992
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Marks 2
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In an undirected connected planar graph G, there are eight vertices and five faces. The number of edges in G is ______
GATE CSE 2021 Set 1
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Let G be a group order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is co...
GATE CSE 2021 Set 1
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An articulation point in a connected graph is a vertex such that removing the vertex and its incident edges disconnects ...
GATE CSE 2021 Set 1
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Graph G is obtained by adding vertex s to K3,4 and making s adjacent to every vertex of K3,4. The minimum number of colo...
GATE CSE 2020
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In a connected graph, bridge is an edge whose removal disconnects a graph. Which one of the following statements is true...
GATE CSE 2015 Set 2
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A graph is self-complementary if it is isomorphic to its complement. For all self-complementary graphs on $$n$$ vertices...
GATE CSE 2015 Set 2
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Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edg...
GATE CSE 2015 Set 1
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Suppose L = { p, q, r, s, t } is a lattice represented by the following Hasse diagram: For any $$x, y ∈ L$$, not necess...
GATE CSE 2015 Set 1
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If $$G$$ is a forest with $$n$$ vertices and $$k$$ connected components, how many edges does $$G$$ have?
GATE CSE 2014 Set 3
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Consider an undirectional graph $$G$$ where self-loops are not allowed. The vertex set of $$G$$ is $$\left\{ {\left( {i,...
GATE CSE 2014 Set 1
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A cycle on $$n$$ vertices is isomorphic to its complement. The value of $$n$$ is __________.
GATE CSE 2014 Set 2
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Let $$\delta $$ denote the minimum degree of a vertex in a graph. For all planar graphs on $$n$$ vertices with $$\delta ...
GATE CSE 2014 Set 3
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An ordered $$n$$-tuple $$\left( {{d_1},\,\,{d_2},\,....,{d_n}} \right)$$ with $${{d_1} \ge ,\,\,{d_2} \ge .... \ge {d_n}...
GATE CSE 2014 Set 1
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The line graph $$L(G)$$ of a simple graph $$G$$ is defined as follows: $$\,\,\,\,$$There is exactly one vertex $$v(e)$$ ...
GATE CSE 2013
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Let $$G$$ be a complete undirected graph on $$6$$ vertices. If vertices of $$G$$ $$\,\,\,\,$$ are labeled, then the numb...
GATE CSE 2012
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Which of the following graphs is isomorphic to ...
GATE CSE 2012
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The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Whic...
GATE CSE 2010
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A binary tree with $$n>1$$ nodes has $${n_1}$$, $${n_2}$$ and $${n_3}$$ nodes of degree one, two and three respective...
GATE CSE 2008
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A binary tree with $$n>1$$ nodes has $${n_1}$$, $${n_2}$$ and $${n_3}$$ nodes of degree one, two and three respective...
GATE CSE 2008
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$$G$$ is a graph on $$n$$ vertices and $$2n-2$$ edges. The edges of $$G$$ can be partitioned into two edge-disjoint span...
GATE CSE 2008
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$$G$$ is a simple undirected graph. Some vertices of $$G$$ are of odd degree. Add a node $$v$$ to $$G$$ and make it adja...
GATE CSE 2008
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Which of the following statements is true for every planar graph on $$n$$ vertices?
GATE CSE 2008
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Let Graph$$(x)$$ be a predicate which denotes that $$x$$ is a graph. Let Connected$$(x)$$ be a predicate which denotes t...
GATE CSE 2007
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Which of the following graphs has an Eulerian circuit?
GATE CSE 2007
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Consider the undirected graph $$G$$ defined as follows. The vertices of $$G$$ are bit strings of length $$n$$. We have a...
GATE CSE 2006
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The $${2^n}$$ vertices of a graph $$G$$ correspond to all subsets of a set of size $$n$$, for $$n \ge 6$$. Two vertices ...
GATE CSE 2006
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The $${2^n}$$ vertices of a graph $$G$$ correspond to all subsets of a set of size $$n$$, for $$n \ge 6$$. Two vertices ...
GATE CSE 2006
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Which one of the following graphs is NOT planar? ...
GATE CSE 2005
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What is the number of vertices in an undirected connected graph with $$27$$ edges, $$6$$ vertices of degree $$2$$, $$\,\...
GATE CSE 2004
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Let $${G_1} = \left( {V,\,{E_1}} \right)$$ and $${G_2} = \left( {V,\,{E_2}} \right)$$ be connected graphs on the same ve...
GATE CSE 2004
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How many graphs on $$n$$ labeled vertices exist which have at least $$\left( {{n^2} - 3n} \right)/2\,\,\,$$ edges?
GATE CSE 2004
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The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned th...
GATE CSE 2004
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How many perfect matchings are there in a complete graph of 6 vertices?
GATE CSE 2003
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$$A$$ graph $$G$$ $$=$$ $$(V, E)$$ satisfies $$\left| E \right| \le \,3\left| V \right| - 6.$$ The min-degree of $$G$$ ...
GATE CSE 2003
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how many undirected graphs (not necessarily connected) can be constructed out of a given $$\,\,\,\,V = \left\{ {{v_1},\,...
GATE CSE 2001
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Prove that in a finite graph, the number of vertices of odd degree is always even.
GATE CSE 1995
GO TO QUESTION
Maximum number of edges in a planar graph with $$n$$ vertices is _______ .
GATE CSE 1992
GO TO QUESTION
The maximum number of possible edges in an undirected graph with a vertices and $$k$$ components is _________ .
GATE CSE 1991
GO TO QUESTION
A graph is planar if and only if,
GATE CSE 1990
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Which of the following graphs is / are planar? (see fig.) ...
GATE CSE 1989
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Marks 5
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Let $$G$$ be a connected, undirected graph. A $$cut$$ in $$G$$ is a set of edges whose removal results in $$G$$ being br...
GATE CSE 1999
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How many minimum spanning tress does the following graph have? Draw them (Weights are assigned to the edges). ...
GATE CSE 1995
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