walks

                                          Walks

Walk is a continuous sequence of vertices and edges starting with a vertex and ending with another vertex with the possible repetition of edges or vertices or both.

An example for a walk 

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                            In a walk if neither vertices nor edges are repeated,it is called a path.


                                                        Here no vertices or edges are repeated .so this is a path.

y-w-u-v-w-x::::this is not a path,it is a walk.
length of a walk or path:-Number of edges in a path or a walk is called the length of the path or the walk.

CIRCUIT

Circuit is a continous sequence of vertices and edges starting and ending at the same vertex,with the possible repetition of edges or vertices.because neither the vertices or nor the edges are repeated the circuit becomes cycle.

  

                                                            EULER GRAPH

In a circuit in a graph were all the edges of the graph are used once and only once.In such cases the graph is called as Euler graph.Every cycle is a Euler circuit.All cycles are circuits but all circuits are not cycles. A graph is said to be an Euiler graph if and only if every vertex is of even degree.

                                                                   EULER WALK
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                               It is a walk in a graph were all the edges of the graph are used once and only once.A graph is said to have an Euler path if and only if it has exactly 2 vertices of odd degree.
                               A graph is said to have an euler path if and only if it has got exactly 2 vertices of odd degree.


NOTE      Who is Euler?

Leonhard Euler(born, local  15 April 1707 – 18 September 1783) was a pioneering Swissmathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work inmechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the pre-eminent mathematician of the 18th century, and one of the greatest mathematicians ever. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes.A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."