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Problem StatementLet 'G' be a connected planar graph with 20 vertices and the degree of each vertex is 3. Find the number of regions in the graph.SolutionBy the sum of degrees theorem, 20 ∑ i=1 deg(Vi) = 2|E|20(3) = 2|E||E| = 30By Euler’s formula,|V| + |R| = |E| + 220+ |R| = 30 + 2|R| = 12Hence, the number of regions is 12.
For this, use the concept of ZEROFILL. It pads the displayed value of the field with zeros up to the display width set in the column definitionLet us first create a table −mysql> create table DemoTable626 (Value int(5) zerofill); Query OK, 0 rows affected (0.71 sec)Insert some records in the table using insert command −mysql> insert into DemoTable626 values(9); Query OK, 1 row affected (0.12 sec) mysql> insert into DemoTable626 values(12); Query OK, 1 row affected (0.13 sec) mysql> insert into DemoTable626 values(567); Query OK, 1 row affected (0.21 sec) mysql> insert into DemoTable626 values(3478); Query OK, 1 row affected ... Read More
Problem StatementHow many simple non-isomorphic graphs are possible with 3 vertices?SolutionThere are 4 non-isomorphic graphs possible with 3 vertices. They are shown below.
Problem StatementWhat is the matching number for the following graph?SolutionNumber of vertices = 9We can match only 8 vertices.Matching number is 4.
Problem StatementWhat is the line covering number for the following graph?SolutionNumber of vertices = |V| = n = 7Line covering number = (α1) ≥ ⌈ n / 2 ⌉ = 3α1 ≥ 3By using 3 edges, we can cover all the vertices.Hence, the line covering number is 3.
Let us first create a table −mysql> create table DemoTable625 ( StudentId int NOT NULL AUTO_INCREMENT PRIMARY KEY, StudentFirstName varchar(100), StudentScore int ); Query OK, 0 rows affected (1.01 sec)Insert some records in the table using insert command −mysql> insert into DemoTable625(StudentFirstName, StudentScore) values('John', 98); Query OK, 1 row affected (0.14 sec) mysql> insert into DemoTable625(StudentFirstName, StudentScore) values('Chris', 39); Query OK, 1 row affected (0.20 sec) mysql> insert into DemoTable625(StudentFirstName, StudentScore) values('Bob', 41); Query OK, 1 row affected (0.11 sec) mysql> insert into DemoTable625(StudentFirstName, StudentScore) values('David', 40); Query OK, 1 row affected (0.14 sec) mysql> insert into DemoTable625(StudentFirstName, StudentScore) ... Read More
Problem StatementWhat is the chromatic number of complete graph Kn?SolutionIn a complete graph, each vertex is adjacent to is remaining (n–1) vertices. Hence, each vertex requires a new color. Hence the chromatic number Kn = n.
A graph is traversable if you can draw a path between all the vertices without retracing the same path. Based on this path, there are some categories like Euler’s path and Euler’s circuit which are described in this chapter.Euler’s PathAn Euler’s path contains each edge of ‘G’ exactly once and each vertex of ‘G’ at least once. A connected graph G is said to be traversable if it contains an Euler’s path.ExampleEuler’s Path = d-c-a-b-d-e.Euler’s CircuitIn an Euler’s path, if the starting vertex is same as its ending vertex, then it is called an Euler’s circuit.ExampleEuler’s Path = a-b-c-d-a-g-f-e-c-a.Euler’s Circuit ... Read More
For this, you can use time_format() and time_diff(). To find the time difference, you need to use the time_diff() method. Let us first create a table −mysql> create table DemoTable624 (PunchIn datetime, PunchOut datetime); Query OK, 0 rows affected (0.60 sec)Insert some records in the table using insert command −mysql> insert into DemoTable624 values('2019-07-14 12:10:00', '2019-07-14 12:50:00'); Query OK, 1 row affected (0.12 sec) mysql> insert into DemoTable624 values('2019-07-14 11:00:00', '2019-07-14 11:30:00'); Query OK, 1 row affected (0.34 sec)Display all records from the table using select statement −mysql> select *from DemoTable624;This will produce the following output −+---------------------+---------------------+ | PunchIn ... Read More
A graph is a set of points, called nodes or vertices, which are interconnected by a set of lines called edges. The study of graphs, or graph theory is an important part of a number of disciplines in the fields of mathematics, engineering and computer science.Graph TheoryDefinition − A graph (denoted as G = (V, E)) consists of a non-empty set of vertices or nodes V and a set of edges E. A vertex a represents an endpoint of an edge. An edge joins two vertices a, b and is represented by set of vertices it connects.Example − Let us ... Read More