Q1: What is a graph in discrete mathematics?
ANSWER A graph G = (V, E) is a mathematical structure consisting of a set of
vertices (V) and a set of edges (E) that connect pairs of vertices.
Q2: What is the difference between a directed graph and an undirected
graph?
ANSWER In an undirected graph, edges have no direction and connect two
vertices symmetrically. In a directed graph (digraph), edges have a direction
indicated by an arrow from one vertex to another.
Q3: What is the degree of a vertex?
ANSWER The degree of a vertex is the number of edges incident to it. In a
directed graph, in-degree is the number of edges pointing to the vertex and out-
degree is the number of edges pointing away.
Q4: State the Handshaking Lemma.
ANSWER The sum of the degrees of all vertices in a graph equals twice the
number of edges: Σdeg(v) = 2|E|. This means every graph has an even number
of vertices with odd degree.
Q5: What is a simple graph?
ANSWER A simple graph is a graph with no self-loops (edges from a vertex to
itself) and no multiple edges (more than one edge between the same pair of
vertices).
Q6: What is a complete graph Kₙ?
ANSWER A complete graph Kₙ is a simple graph where every pair of distinct
vertices is connected by exactly one edge. It has n(n-1)/2 edges.
Q7: What is a bipartite graph?
ANSWER A bipartite graph is a graph whose vertices can be divided into two
disjoint sets U and V such that every edge connects a vertex in U to a vertex in
V. No edges exist within the same set.
,Q8: What is a subgraph?
ANSWER A subgraph H of graph G is a graph formed by a subset of the
vertices and edges of G, where every edge in H connects two vertices that are
also in H.
Q9: What is a path in a graph?
ANSWER A path is a sequence of vertices v₁, v₂, ..., vₙ such that consecutive
vertices are connected by edges, and no vertex is repeated (in a simple path).
Q10: What is a cycle in a graph?
ANSWER A cycle is a closed path that starts and ends at the same vertex,
visiting each vertex and edge exactly once (except the starting/ending vertex).
Q11: What is graph connectivity?
ANSWER A graph is connected if there exists a path between every pair of
vertices. A graph that is not connected consists of two or more connected
components.
Q12: What is a tree in graph theory?
ANSWER A tree is a connected, acyclic undirected graph. A tree with n vertices
always has exactly n-1 edges.
Q13: What is a spanning tree?
ANSWER A spanning tree of a graph G is a subgraph that includes all vertices
of G and is a tree. Every connected graph has at least one spanning tree.
Q14: What is Euler's circuit theorem?
ANSWER An Eulerian circuit (a circuit that traverses every edge exactly once)
exists in a connected graph if and only if every vertex has even degree.
Q15: What is an Eulerian path?
ANSWER An Eulerian path traverses every edge exactly once. It exists in a
connected graph if and only if there are exactly 0 or 2 vertices of odd degree.
Q16: What is a Hamiltonian circuit?
ANSWER A Hamiltonian circuit is a circuit that visits every vertex exactly once
and returns to the starting vertex. Unlike Euler, there is no simple necessary and
sufficient condition for their existence.
Q17: What is the adjacency matrix of a graph?
ANSWER An adjacency matrix is an n×n matrix A where A[i][j] = 1 if there is an
edge between vertices i and j, and 0 otherwise. For undirected graphs, it is
symmetric.
Q18: What is the incidence matrix of a graph?
, ANSWER An incidence matrix is an n×m matrix where n is the number of
vertices and m the number of edges. Entry [i][j] = 1 if vertex i is incident to edge j,
0 otherwise.
Q19: What is graph isomorphism?
ANSWER Two graphs G and H are isomorphic if there exists a bijection
between their vertex sets that preserves adjacency. They have the same
structure but possibly different labels.
Q20: What is a planar graph?
ANSWER A planar graph is a graph that can be drawn in a plane without any
edges crossing. K₄ is planar, while K₅ and K₃,₃ are not.
Q21: State Euler's formula for planar graphs.
ANSWER For a connected planar graph: v - e + f = 2, where v = number of
vertices, e = number of edges, and f = number of faces (including the outer
infinite face).
Q22: What is graph coloring?
ANSWER Graph coloring assigns colors to vertices such that no two adjacent
vertices share the same color. The chromatic number χ(G) is the minimum
number of colors needed.
Q23: What is the Four Color Theorem?
ANSWER Every planar graph can be colored with at most four colors such that
no two adjacent vertices share the same color. Proved in 1976 by Appel and
Haken.
Q24: What is a weighted graph?
ANSWER A weighted graph is a graph where each edge is assigned a
numerical weight or cost. These are used in shortest path and minimum
spanning tree algorithms.
Q25: What is Dijkstra's algorithm used for?
ANSWER Dijkstra's algorithm finds the shortest path between a source vertex
and all other vertices in a weighted graph with non-negative edge weights. It runs
in O((V+E)log V) with a priority queue.
Q26: What is Kruskal's algorithm?
ANSWER Kruskal's algorithm finds a minimum spanning tree by sorting all
edges by weight and greedily adding edges that don't create a cycle. Uses a
Union-Find data structure.
Q27: What is Prim's algorithm?
ANSWER A graph G = (V, E) is a mathematical structure consisting of a set of
vertices (V) and a set of edges (E) that connect pairs of vertices.
Q2: What is the difference between a directed graph and an undirected
graph?
ANSWER In an undirected graph, edges have no direction and connect two
vertices symmetrically. In a directed graph (digraph), edges have a direction
indicated by an arrow from one vertex to another.
Q3: What is the degree of a vertex?
ANSWER The degree of a vertex is the number of edges incident to it. In a
directed graph, in-degree is the number of edges pointing to the vertex and out-
degree is the number of edges pointing away.
Q4: State the Handshaking Lemma.
ANSWER The sum of the degrees of all vertices in a graph equals twice the
number of edges: Σdeg(v) = 2|E|. This means every graph has an even number
of vertices with odd degree.
Q5: What is a simple graph?
ANSWER A simple graph is a graph with no self-loops (edges from a vertex to
itself) and no multiple edges (more than one edge between the same pair of
vertices).
Q6: What is a complete graph Kₙ?
ANSWER A complete graph Kₙ is a simple graph where every pair of distinct
vertices is connected by exactly one edge. It has n(n-1)/2 edges.
Q7: What is a bipartite graph?
ANSWER A bipartite graph is a graph whose vertices can be divided into two
disjoint sets U and V such that every edge connects a vertex in U to a vertex in
V. No edges exist within the same set.
,Q8: What is a subgraph?
ANSWER A subgraph H of graph G is a graph formed by a subset of the
vertices and edges of G, where every edge in H connects two vertices that are
also in H.
Q9: What is a path in a graph?
ANSWER A path is a sequence of vertices v₁, v₂, ..., vₙ such that consecutive
vertices are connected by edges, and no vertex is repeated (in a simple path).
Q10: What is a cycle in a graph?
ANSWER A cycle is a closed path that starts and ends at the same vertex,
visiting each vertex and edge exactly once (except the starting/ending vertex).
Q11: What is graph connectivity?
ANSWER A graph is connected if there exists a path between every pair of
vertices. A graph that is not connected consists of two or more connected
components.
Q12: What is a tree in graph theory?
ANSWER A tree is a connected, acyclic undirected graph. A tree with n vertices
always has exactly n-1 edges.
Q13: What is a spanning tree?
ANSWER A spanning tree of a graph G is a subgraph that includes all vertices
of G and is a tree. Every connected graph has at least one spanning tree.
Q14: What is Euler's circuit theorem?
ANSWER An Eulerian circuit (a circuit that traverses every edge exactly once)
exists in a connected graph if and only if every vertex has even degree.
Q15: What is an Eulerian path?
ANSWER An Eulerian path traverses every edge exactly once. It exists in a
connected graph if and only if there are exactly 0 or 2 vertices of odd degree.
Q16: What is a Hamiltonian circuit?
ANSWER A Hamiltonian circuit is a circuit that visits every vertex exactly once
and returns to the starting vertex. Unlike Euler, there is no simple necessary and
sufficient condition for their existence.
Q17: What is the adjacency matrix of a graph?
ANSWER An adjacency matrix is an n×n matrix A where A[i][j] = 1 if there is an
edge between vertices i and j, and 0 otherwise. For undirected graphs, it is
symmetric.
Q18: What is the incidence matrix of a graph?
, ANSWER An incidence matrix is an n×m matrix where n is the number of
vertices and m the number of edges. Entry [i][j] = 1 if vertex i is incident to edge j,
0 otherwise.
Q19: What is graph isomorphism?
ANSWER Two graphs G and H are isomorphic if there exists a bijection
between their vertex sets that preserves adjacency. They have the same
structure but possibly different labels.
Q20: What is a planar graph?
ANSWER A planar graph is a graph that can be drawn in a plane without any
edges crossing. K₄ is planar, while K₅ and K₃,₃ are not.
Q21: State Euler's formula for planar graphs.
ANSWER For a connected planar graph: v - e + f = 2, where v = number of
vertices, e = number of edges, and f = number of faces (including the outer
infinite face).
Q22: What is graph coloring?
ANSWER Graph coloring assigns colors to vertices such that no two adjacent
vertices share the same color. The chromatic number χ(G) is the minimum
number of colors needed.
Q23: What is the Four Color Theorem?
ANSWER Every planar graph can be colored with at most four colors such that
no two adjacent vertices share the same color. Proved in 1976 by Appel and
Haken.
Q24: What is a weighted graph?
ANSWER A weighted graph is a graph where each edge is assigned a
numerical weight or cost. These are used in shortest path and minimum
spanning tree algorithms.
Q25: What is Dijkstra's algorithm used for?
ANSWER Dijkstra's algorithm finds the shortest path between a source vertex
and all other vertices in a weighted graph with non-negative edge weights. It runs
in O((V+E)log V) with a priority queue.
Q26: What is Kruskal's algorithm?
ANSWER Kruskal's algorithm finds a minimum spanning tree by sorting all
edges by weight and greedily adding edges that don't create a cycle. Uses a
Union-Find data structure.
Q27: What is Prim's algorithm?
