The number of edges in a regular graph of degree d and n vertices is _________.

The number of distinct simple graphs with upto three nodes is

Prove that in finite graph, the number of vertices of odd degree is always even.

The minimum number of edges in a connected cyclic graph on n vertices is:

Graph G is obtained by adding vertex s to K_3,4 and making s adjacent to every vertex of K_3,4. The minimum number of colours required to edge-colour G is _____.

Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to

Which of the following graphs is isomorphic to

Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on the plane is equal to

Let G be a graph with 100 vertices numbered 1 to 100. Two vertices i and j are adjacent if |i - j| = 8 or |i - j| = 12. The number of connected components in G is

(a) Prove by induction that the expression for the number of diagonals in a polygon of n sides is n(n-3)/2.

(b) Let R be a binary relation on A = {a, b, c, d, e, f, g, h} represented by following two component digraph. Find the smallest integers m and n such that mm = Rⁿ.

Let (A, *) be a semi group. Furthermore, for every a and b in A, if a ≠ b, then a*b ≠ b*a.

(a) Show that for every a in A

 a*a = a 

(b) Show that for every a, b in A

 a*b*a = a  

(c) Show that for every a, b, c in A

 a*b*c = a*c 

Let G be a connected, undirected graph. A cut in G is a set of edges whose removal results in 0 being broken into two or more components which are not connected with each other. The size of a cut is called its cardinality. A men-cut of G is a cut in G of minimum cardinality. Consider the following graph.

(a) Which of the following sets of edges is a cut?
(i) {(A,B), (E,F), (B,D), (A,E), (A,D)}
(ii) {(B,D), (C,F), (A,B)}

(b) What is the cardinality of a min-cut in the graph?

(c) Prove that if a connected undirected graph G with n vertices has a min-cut of cardinality K, then G has atleast (nk/2) edges.

Let S be a set of n elements {1, 2, …., n} and G a graph with 2ⁿ vertices, each vertex corresponding to a distinct subset of S. Two vertices are adjacent iff the symmetric difference of the corresponding sets has exactly 2 elements. Note: The symmetric difference of two sets R¹ and R² is defined as (R¹/R²)∪(R²/R¹)
(a) Every vertex in G has the same degree. What is the degree of a vertex in G?
(b) How many connected components does G have?

How many undirected graphs (not necessarily connected) can be constructed out of a given set V = {v₁, v₂, ...,v_n} of n vertices?

Maximum number of edges in a n-node undirected graph without self loops is

The minimum number of colours required to colour the vertices of a cycle with η nodes in such a way that no two adjacent nodes have the same colour is

A graph G = (V,E) satisfies |E|≤ 3|V|-6. The min-degree of G is defined as . Therefore, min-degree of G cannot be

How many perfect matching are there in a complete graph of 6 vertices ?

Let G be an arbitrary graph with n nodes and k components. If a vertex is removed from G, the number of components in the resultant graph must necessarily lie between

(a) If G is a group of even order, then show that there exists an element a ≠ e, the identity in g, such that a² = e

(b) Consider the set of integers {1,2,3, 4,6,8,12,24} together with the two binary operations LCM (lowest common multiple) and GCD (greatest common divisor). Which of the following algebraic structures does this represent?

 (i) group            (ii) ring
 (iii) field          (iv) lattice  

Let T be a Depth First Tree of a undirected graph G. An array P indexed by vertices of G is given. P[V] is the parent of vertex V, in T. Parent of the root is the root itself.

Give a method for finding and printing the cycle formed if the edge (u,v) of G not in T (i.e., e ∈ G − T) is now added to T.

Time taken by your method must be proportional to the length of the cycle.

Describe the algorithm in a PASCAL – like language. Assume that the variables have been suitably declared.

A non-planar graph with minimum number of vertices has

Maximum number of edges in a planar graph with n vertices is ________

The maximum number of possible edges in an undirected graph with a vertices and k components is ________.

What is the number of vertices in an undirected connected graph with 27 edges, 6 vertices of degree 2, 3 vertices of degree 4 and remaining of degree 3?

What is the maximum number of edges in an acyclic undirected graph with n vertices?

Let G be a directed graph whose vertex set is the set of numbers from 1 to 100. There is an edge from a vertex i to a vertex j iff either j = i + 1 or j = 3i. The minimum number of edges in a path in G from vertex 1 to vertex 100 is

Let G be a weighted undirected graph and e be an edge with maximum weight in G. Suppose there is a minimum weight spanning tree in G containing the edge e. Which of the following statements is always TRUE?

Consider the undirected graph G defined as follows. The vertices of G are bit strings of length n. We have an edge between vertex u and vertex v if and only if u and v differ in exactly one bit position (in other words, v can be obtained from u by flipping a single bit). The ratio of the chromatic number of G to the diameter of G is

G is a simple undirected graph. Some vertices of G are of odd degree. Add a node v to G and make it adjacent to each odd degree vertex of G. The resultant graph is sure to be

What is the size of the smallest MIS(Maximal Independent Set) of a chain of nine nodes?

What is the chromatic number of the following graph?

Which of the following graphs is/are planner?

Choose the correct alternatives (More than one may be correct).

Let G₁ = (V,E₁) and G₂ = (V,E₂) be connected graphs on the same vertex set V with more than two vertices. If G₁ ∩ G₂ = (V, E₁ ∩ E₂) is not a connected graph, then the graph G₁ U G₂ = (V, E₁ U E₂)

How many graphs on n labeled vertices exist which have at least (n² - 3n)/2 edges?

The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same colour, is

If G is a forest with n vertices and k connected components, how many edges does G have?

Let δ denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with δ ≥ 3, which one of the following is TRUE?

There are two elements x, y in a group (G,∗) such that every element in the group can be written as a product of some number of x's and y's in some order. It is known that

  x * x = y * y = x * y * x = y * x * y * x = e   

where e is the identity element. The maximum number of elements in such a group is ________.

A graph is self-complementary if it is isomorphic to its complement for all self-complementary graphs on n vertices, n is

In a connected graph, a bridge is an edge whose removal disconnects a graph. Which one of the following statements is true?

Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is _______________.

The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.

G is an undirected graph with n vertices and 25 edges such that each vertex of G has degree at least 3. Then the maximum possible value of n is ___________.

Let G be a graph with 100! vertices, with each vertex labeled by a distinct permutation of the numbers 1, 2, …, 100. There is an edge between vertices u and v if and only if the label of u can be obtained by swapping two adjacent numbers in the label of v. Let y denote the degree of a vertex in G, and z denote the number of connected components in G.

Then, y + 10z = ___________.

The chromatic number of the following graph is _______.

Let G be an undirected complete graph on n vertices, where n > 2. Then, the number of different Hamiltonian cycles in G is equal to

Consider an undirected graph where self-loops are not allowed. The vertex set of G is {(i,j): 1 ≤ i ≤ 12, 1 ≤ j ≤ 12}. There is an edge between (a,b) and (c,d) if |a - c| ≤ 1 and |b - d| ≤ 1. The number of edges in this graph is __________.

An ordered -tuple (d₁, d₂, ..., d_n) with d₁ ≥ d₂ ≥ ... d_n is called graphic if there exists a simple undirected graph with n vertices having degrees d₁, d₂, ..., d_n respectively. Which of the following 6-tuples is NOT graphic?

Let G = (V,E) be a directed graph where V is the set of vertices and E the set of edges. Then which one of the following graphs has the same strongly connected components as G?

Which of the following statements is true for every planar graph on n vertices?