GATE CSE 2008 | Graph Theory Question 44 | Discrete Mathematics

GATE CSE 2008

MCQ (Single Correct Answer)

-0.6

A binary tree with $$n>1$$ nodes has $${n_1}$$, $${n_2}$$ and $${n_3}$$ nodes of degree one, two and three respectively. The degree of a node is defined as the number of its neighbours.

$${n_3}$$ can be expressed as:

$${n_1}$$ $$+$$ $${n_1}$$ $$-$$ $$1$$

$${n_1}$$ $$-$$ $$2$$

$$\left[ {{{{n_1} + {n_2}} \over 2}} \right]$$

$${n_2}$$ $$-$$ $$1$$

GATE CSE 2008

MCQ (Single Correct Answer)

-0.6

A binary tree with $$n>1$$ nodes has $${n_1}$$, $${n_2}$$ and $${n_3}$$ nodes of degree one, two and three respectively. The degree of a node is defined as the number of its neighbours.

Starting with the above tree, while there remains a node $$v$$ of degree two in the tree, add an edge between the two neighbours of $$v$$ and then remove $$v$$ from the tree. How many edges will remain at the end of the process?

$${2^ * }{n_1} - 3$$

$${n_2} + {2^ * }{n_1} - 2$$

$${n_3} - {n_2}$$

$${n_2} + {n_1} - 2$$

GATE CSE 2007

MCQ (Single Correct Answer)

-0.6

Which of the following graphs has an Eulerian circuit?

Any $$k$$-regular graph where $$k$$ is an even number.

A complete graph on 90 vertices.

The complement of a cycle on 25 vertices.

None of the above.

GATE CSE 2007

MCQ (Single Correct Answer)

-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: $$Not\,\,\,every\,\,\,graph\,\,\,is\,\,\,connected?$$

$$\neg \forall x\left( {Graph\left( x \right) \Rightarrow Connected\left( x \right)} \right)$$

$$\exists x\left( {Graph\left( x \right) \wedge \neg Connected\left( x \right)} \right)$$

$$\neg \forall x\left( {\neg Graph\left( x \right) \vee Connected\left( x \right)} \right)$$

$$\forall x\left( {Graph\left( x \right) \Rightarrow \neg Connected\left( x \right)} \right)$$

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