Theorem (Sum of Degrees of Vertices Theorem):
Suppose a graph has n vertices with degrees d ₁ , d ₂ , d ₃ , ..., d _n.
Add together all degrees to get a new number
d ₁ + d ₂ + d ₃ + . .. + d _n = D _v . Then D _v = 2 e .
In words, for any graph the sum of the degrees of the vertices equals twice the number of edges.

Adjacency Matrix
● Uses O(n​ ²​ ) memory
● It fast to lookup and check for presence or absence of a specific edge between any two nodes O(1)
● It slow to iterate over all edges
● It slow to add/delete a node is a complex operation O(n​ ²​ )
Adjacency List
● Use memory pending on the number of edges (not number of nodes), Which might save a lot of memory if the adjacency matrix is a sparse
● Finding the presence or absence specific edge between any two nodes Is slightly slower than with the matrix O(k) where k number of neighbors nodes
● It is fast to iterate over all edges,because you can access any node neighbors directly
● It is fast to add/delete a node is easier than the matrix representation

● Hamiltonian path, also called a Hamilton path, is a graph path between two vertices of a graph that visits each vertex exactly once. If a Hamiltonian path exists whose endpoints are adjacent, then the resulting graph cycle is called a Hamiltonian cycle (or Hamiltonian cycle)
● An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once.
● An Euler path starts and ends at different vertices.
● An Euler circuit starts and ends at the same vertex.

● Depth First Search (DFS) algorithm traverses a graph in a depthward motion and uses a stack to remember to get the next vertex to start a search, when a dead end occurs in any iteration.
● Breadth-first search (BFS) is an algorithm for traversing or searching tree or graph data structures. It starts at the tree root (or some arbitrary node of a graph, sometimes referred to as a 'search key'), and explores all of the neighbor nodes at the present depth prior to moving on to the nodes at the next depth level.

The basic idea of DFS is
● Pick a starting node and push all its adjacent nodes into a stack.
● Pop a node from stack to select the next node to visit and push all its adjacent nodes into a stack.
● Repeat this process until the stack is empty. However, ensure that the nodes that are visited are marked.
● This will prevent you from visiting the same node more than once. If you do not mark the nodes that are visited and you visit the same node more than once, you may end up in an infinite loop.

● A bridge ,cut-edge, or cut arc is an edge of a graph whose deletion increases its number of connected components.
● Equivalently, an edge is a bridge if and only if it is not contained in any cycle.
● The removal of edges (a,b) and (e,f) makes graph disconnected.

An Euler circuit in a graph G is a simple circuit containing every edge of G exactly once
An Euler path in G is a simple path containing every edge of G exactly once.
An Euler path starts and ends at different vertices.
An Euler circuit starts and ends at the same vertex.

Use Euler's formula ,V−E+R=2
Where V is the number of vertices, E is the number of edges, and R is the number of regions.
R=2-V+E=2-8+13=7

● The collection of all subsets of a set A is called the power set of A.
● The empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero
● Common notations for the empty set include "{}", " ", and "∅".

● A Cartesian product is a ​ mathematical operation​ that returns a ​ set​ (or product set or simply product) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ​ ordered pairs​ (a, b) where a ∈ A and b ∈ B.
● In the question, Set A consists of two elements and Set B consists of two elements. So the total ordered pairs are four.
● Each ordered pair consists of one element from Set-A and another element from Set-B.

The cardinality of a set is defined as the total number of distinct items in that set and power set is defined as the set of all subsets of a set.
Power set consists of ​ {}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}
So power set consists of all unique items then cardinality is 8

● Use Euler's formula ,V−E+R=2
● Where V is the number of vertices, E is the number of edges, and R is the number of regions.
● R=2-V+E=2-13+19=8
● We can also term region as face

One simple method is, by using truth table we can find the statement is true or not.

The last two columns of the above table are different. So option A is false.

The number of bus lines between a and b =4
The number of bus lines between b and c =3
The number of possible combinations between a and c going through b are 4 * 3 = 12.
if you're talking about round trip, he has 12 possible ways to get there and 12 possible ways to get back, so the total possible ways is 12 * 12 = 144.

Octagon consists of the 8 vertices and we can draw 5 diagonals

So, we can construct 5*8 = 40 diagonals.
But we have constructed each diagonal twice, once from each of its ends. Thus there are 20 diagonals in a regular octagon.