1
GATE CSE 2007
+2
-0.6
Which of the following graphs has an Eulerian circuit?
A
Any $$k$$-regular graph where $$k$$ is an even number.
B
A complete graph on 90 vertices.
C
The complement of a cycle on 25 vertices.
D
None of the above.
2
GATE CSE 2007
+2
-0.6
Let Graph$$(x)$$ be a predicate which denotes that $$x$$ is a graph. Let Connected$$(x)$$ be a predicate which denotes that $$x$$ is connected. Which of the following first order logic sentences DOES NOT represent the statement: $$Not\,\,\,every\,\,\,graph\,\,\,is\,\,\,connected?$$
A
$$\neg \forall x\left( {Graph\left( x \right) \Rightarrow Connected\left( x \right)} \right)$$
B
$$\exists x\left( {Graph\left( x \right) \wedge \neg Connected\left( x \right)} \right)$$
C
$$\neg \forall x\left( {\neg Graph\left( x \right) \vee Connected\left( x \right)} \right)$$
D
$$\forall x\left( {Graph\left( x \right) \Rightarrow \neg Connected\left( x \right)} \right)$$
3
GATE CSE 2006
+2
-0.6
The $${2^n}$$ vertices of a graph $$G$$ correspond to all subsets of a set of size $$n$$, for $$n \ge 6$$. Two vertices of $$G$$ are adjacent if and only if the corresponding sets intersect in exactly two elements.

the number of vertices of degree zero in $$G$$ is

A
$$1$$
B
$$n$$
C
$$n + 1$$
D
$${2^n}$$
4
GATE CSE 2006
+2
-0.6
The $${2^n}$$ vertices of a graph $$G$$ correspond to all subsets of a set of size $$n$$, for $$n \ge 6$$. Two vertices of $$G$$ are adjacent if and only if the corresponding sets intersect in exactly two elements.

The maximum degree of a vertex in $$G$$ is

A
$$\left( {\mathop 2\limits^{n/2} } \right){2^{n/2}}$$
B
$${2^{n - 2}}$$
C
$${2^{n - 3}} \times 3$$
D
$${2^{n - 1}}$$
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