MATHS_6/10-GRAPHS : MATHS_6/10-GRAPHS BY B. MANJULA
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GRAPHS : GRAPHS A set of points together with line segments joining in pairs is called a network or graph
The points in a graph are called the nodes in the graph
The line segments joining the nodes in pairs are called arcs in the graph
A point which is not named is taken as a node
The arc in a graph which are line segments may be straight or curved.
GRAPHS, ARCS, NODES : GRAPHS, ARCS, NODES In the figure A B C D and E are nodes.
AB, BC ,CD ,AE AND ED are nine arcs. Here there are five nodes ie., N = 5 and 6 arcs ie., A = 6
Number of sides are denoted by N and number of Arcs by A
NETWORKS : NETWORKS A net work is said to be traversable if it can be traced without lifting the pencil from the paper and without retracing any arc from start to finish in one sweep, but it is permitted to pass through any node any number of times.
In a graph the number of arcs form a mode is called the order of the node called 1,2,3 node
A single arc connecting a node to itself is called a loop at the node.
A loop can be traced in a clockwise as well as in the anti-clockwise, directions, the order of a node with a loop is increased by 2
EULER’S SOLUTIONS : EULER’S SOLUTIONS A graph is traversable (1) if it has only even nodes. (2) If its has two odd nodes
A graph is not traversable if it has more than two odd nodes
If the graph has two odd notes it can be traversed only if we start from an odd node and not otherwise and end with the other odd node
MATRIX OF A GRAPH : MATRIX OF A GRAPH The information regarding the number of arcs in a graph connecting the nodes can be displayed by a matrix, which is called network matrix
The matrix which gives the number of arcs in the given graph is called matrix of the graph or network matrix
The number of rows or columns of the matrix of a graph is the number of nodes
The sum of number of row or column is the order of corresponding node
The sum of the numbers in the matrix is the sum of the orders in the nodes which is equal to 2%(total no of arcs in the graph
EULER’S FORMULAE : EULER’S FORMULAE FOR GRAPHS
Any area bounded by arcs is called Region. It is denoted by the letter ‘R’
In a graph, if N is the number of nodes A is the number of arcs and R is the number of regions into which the graph divides the plane of graph then, N + R = A + 2
A loop is considered in a single arc while counting
EULER’S FORMULAE_CONTD : EULER’S FORMULAE_CONTD FOR POLYHYDRA AND PLYHEDRAL GRAPHS
A polygon is a closed plane figure bounded by straight line. Segments are called sides of polygons and region bounded by a polygon is called Polygonal region
A polygon having equal sides and equal angles is called Regular Polygon and can be inscribed in a circle
A closed figure in three dimensional space bounded by polygon is called polyhedron
EULER’S FORMULAE_CONTD : EULER’S FORMULAE_CONTD A solid bounded by a polyhedron is called Polyhedral solid. For a polyhedron , the number of faces is denoted by F and vertices by V.
In a Polyhedron if its faces are congruent regular polygons then it is called regular polyhedron.
The regular polyhedra are Tetrahedron, Triangular base pyramid, Hexahedron(Cube), Octohedron, Dodecahedron and Icosahedron
CONCLUSION : CONCLUSION PROBLEMS WILL BE SOVED IN PUBLIC CLASSES ON WIZIQ
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