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# Data Structure MCQ – Multigraph and Hypergraph

Given Adjacency matrices determine which of them are PseudoGraphs?

i) {{1,0} {0,1}}

ii) {{0,1}{1,0}}

iii) {{0,0,1}{0,1,0}{1,0,0}}

a) only i)

b) ii) and iii)

c) i) and iii)

d) i) ii) and iii)

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Answer: c

Explanation: In i) self loops exist for both the vertices, in iii) self loop exists in the second vertex.

All undirected Multigraphs contain eulerian cycles.

a) True

b) False

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Answer: a

Explanation: Only graphs with every vertex having even degree have eulerian circuits or cycles.

Determine the number of vertices for the given Graph or Multigraph?

G is a 4-regular Graph having 12 edges.

a) 3

b) 6

c) 4

d) Information given is insufficient

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Answer: b

Explanation: Sum of degrees of all the edges equal to 2 times the number of edges. 2*12=4*n, n=>6.

Which of the following statement is true.

a) There exists a Simple Graph having 10 vertices such that minimum degree of the graph is 0 and maximum degree is 9

b) There exists a MultiGraph having 10 vertices such that minimum degree of the graph is 0 and maximum degree is 9

c) There exists a MultiGraph as well as a Simple Graph having 10 vertices such that minimum degree of the graph is 0 and maximum degree is 9

d) None of the mentioned

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Answer: b

Explanation: If a vertex has a degree 9 that means it is connected to all the other vertices, in case of Multigraphs for an isolate vertex, and a multiple edge may compensate.

Given Adjacency matrices determine which of them are PseudoGraphs?

i) {{1,0} {0,1}}

ii) {{0,1}{1,0}}

iii) {{0,0,1}{0,1,0}{1,0,0}}

a) only i)

b) ii) and iii)

c) i) and iii)

d) i) ii) and iii)

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Answer: c

Explanation: In i) self loops exist for both the vertices, in iii) self loop exists in the second vertex.

Possible number of labelled simple Directed, Pseudo and Multigarphs exist having 2 vertices?

a) 3, Infinite, 4

b) 4, 3, Infinite

c) 4, Infinite, infinite

d) 4, Infinite, Infinite

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Answer: d

Explanation: MultiGraphs and PseudoGraphs may have infinite number of edges, while 4 possible simple graphs exist.

Which of the following is a HyperGraph, where V is the set of vertices, E is the set of edges?

a) V = {v1, v2, v3} E = {e1, e2} = {{v2, v3} {v1, v3}}

b) V = {v1, v2} E = {e1} = {{v1, v2}}

c) V = {v1, v2, v3} E = {e1, e2, e3} = {{v2, v3}{v3, v1}{v2, v1}}

d) All of the mentioned

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Answer: d

Explanation: In a uniform Graph all the hyper-edges have the same cardinality.

What would be the Incidence Matrix of the given HyperGraph?

V = {x,y,z} E = {{x,y}{y}{x,z}{z,y}}

a) {{1,0,1,0},

{1,1,0,1},

{0,0,1,1}}

b) {{1,1,0,0},

{0,1,0,0},

{1,1,1,0}}

c) {{0,1,0,1},

{0,0,1,0},

{1,1,0,0}}

d) None of the Mentioned

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Answer: a

Explanation: The columns represent edges while rows represent vertices.

What is the degree sequence of the given HyperGraph, in non-increasing order.

V = {v1,v2,v3,v4,v5,v6} E = {{v1,v4,v5} {v2,v3,v4,v5} {v2} {v1} {v1,v6}}

a) 3,2,1,1,1,1

b) 3,2,2,2,1,1

c) 3,2,2,2,2,1

d) 3,2,2,1,1,1

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Answer: b

Explanation: The degree of v1,v2,v3,v4,v5,v6 is 3,2,1,2,2,1 respectively.

MultiGraphs having self-loops are called PseudoGraphs?

a) True

b) False

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Answer: a

Explanation: All PsuedoGraphs are MultiGraphs, but all MultiGraphs are not PseudoGraphs as all PseudoGraphs have self loop, but all MultiGraphs do not have self loops.