A complete bipartite graph is a one in which each vertex in set X has an edge with set Y. Let n be the total number of vertices. For maximum number of edges, the total number of vertices hat should be present on set X is?

We can prove this by calculus. Let x be the total number of vertices on set X. Therefore set Y will have n-x. We have to maximize x*(n-x). This is true when x=n/2.

Let D be a simple graph on 10 vertices such that there is a vertex of degree 1, a vertex of degree 2, a vertex of degree 3, a vertex of degree 4, a vertex of degree 5, a vertex of degree 6, a vertex of degree 7, a vertex of degree 8 and a vertex of degree 9. What can be the degree of the last vertex?

The primitive translation vectors of the body centred cubic lattice are $$\overrightarrow {\bf{a}} = \frac{a}{2}\left( {{\bf{\hat x}} + {\bf{\hat y}} - {\bf{\hat z}}} \right),\,\overrightarrow {\bf{b}} = \frac{a}{2}\left( { - {\bf{\hat x}} + {\bf{\hat y}} + {\bf{\hat z}}} \right)$$ and $$\overrightarrow {\bf{c}} = \frac{a}{2}\left( {{\bf{\hat x}} - {\bf{\hat y}} + {\bf{\hat z}}} \right)$$ . The primitive translation vectors $$\overrightarrow {\bf{A}} ,\,\overrightarrow {\bf{B}} $$ and $$\overrightarrow {\bf{C}} $$ of the reciprocal lattice are

$$\overrightarrow {\bf{A}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} - {\bf{\hat y}}} \right);\,\overrightarrow {\bf{B}} = \frac{{2\pi }}{a}\left( {{\bf{\hat y}} + {\bf{\hat z}}} \right);\,\overrightarrow {\bf{C}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} + {\bf{\hat z}}} \right)$$

$$\overrightarrow {\bf{A}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} - {\bf{\hat y}}} \right);\,\overrightarrow {\bf{B}} = \frac{{2\pi }}{a}\left( {{\bf{\hat y}} - {\bf{\hat z}}} \right);\,\overrightarrow {\bf{C}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} + {\bf{\hat z}}} \right)$$

$$\overrightarrow {\bf{A}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} + {\bf{\hat y}}} \right);\,\overrightarrow {\bf{B}} = \frac{{2\pi }}{a}\left( {{\bf{\hat y}} + {\bf{\hat z}}} \right);\,\overrightarrow {\bf{C}} = \frac{{2\pi }}{a}\left( {{\bf{\hat x}} - {\bf{\hat z}}} \right)$$

Let G be a directed graph whose vertex set is the set of numbers from 1 to 50. There is an edge from a vertex i to a vertex j if and only if either j = i + 1 or j = 3i. Calculate the minimum number of edges in a path in G from vertex 1 to vertex 50.

There are four students in a class namely A, B, C and D. A tells that a triangle is a bipartite graph. B tells pentagon is a bipartite graph. C tells square is a bipartite graph. D tells heptagon is a bipartite graph. Who among the following is correct?

A person wants to visit some places. He starts from a vertex and then wants to visit every vertex till it finishes from one vertex, backtracks and then explore other vertex from same vertex. What algorithm he should use?

Depth First Search

Breadth First Search

Trim’s algorithm

Kruskal’s Algorithm

$${\bf{\hat A}}$$ and $${\bf{\hat B}}$$ represent two physical characteristics of a quantum system. If $${\bf{\hat A}}$$ is Hermitian, then for the product $${\bf{\hat A\hat B}}$$ to be Hermitian, it is sufficient that

$${\bf{\hat B}}$$ is Hermitian

$${\bf{\hat B}}$$ is anti-Hermitian

$${\bf{\hat B}}$$ is Hermitian and $${\bf{\hat A}}$$ and $${\bf{\hat B}}$$ commute

$${\bf{\hat B}}$$ is Hermitian and $${\bf{\hat A}}$$ and $${\bf{\hat B}}$$ anti-commute

A k-regular bipartite graph is the one in which degree of each vertices is k for all the vertices in the graph. Given that the bipartitions of this graph are U and V respectively. What is the relation between them?

Number of vertices in U=Number of vertices in V

Number of vertices in U not equal to number of vertices in V

Number of vertices in U always greater than the number of vertices in V

Nothing can be said

An undirected graph has 8 vertices labelled 1, 2, …,8 and 31 edges. Vertices 1, 3, 5, 7 have degree 8 and vertices 2, 4, 6, 8 have degree 7. What is the degree of vertex 8?

An undirected graph G has bit strings of length 100 in its vertices and there is an edge between vertex u and vertex v if and only if u and v differ in exactly one bit position. Determine the ratio of the chromatic number of G to the diameter of G?

1/2101

1/50

1/100

1/20

$${{\bf{\hat A}}}$$ and $${{\bf{\hat B}}}$$ are two quantum mechanical operators. If $$\left[ {{\bf{\hat A}},\,{\bf{\hat B}}} \right

$$ stands for the commutator of $${{\bf{\hat A}}}$$ and $${{\bf{\hat B}}}$$ then $$\left,\,\left} \right

$$ is equal to

<p><span>A.</span> $${\bf{\hat A\hat B\hat A\hat B}} - {\bf{\hat B\hat A\hat B\hat A}}$$

</span> $${\bf{\hat A}}\left( {{\bf{\hat A\hat B}} - {\bf{\hat B\hat A}}} \right) - {\bf{\hat B}}\left( {{\bf{\hat B\hat A}} - {\bf{\hat A\hat B}}} \right)$$