EURO 2020 Fantasy Football - Pupil Rules in Deutsch
Build your EURO 2020 fantasy football team with 5 players from 5 different countries under a 20 million budget. Understand the scoring system for goalkeepers, defenders, midfielders, and strikers to maximize your points. Dive into the competition with clear guidelines and strategies for a thrilling fantasy football experience auf Deutsch!
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Presentation Transcript
Graphs and Trees This handout: Total degree of a graph Applications of Graphs
Graph properties Definition: The total degree of a graph is the sum of the degrees of all its nodes. Theorem: If G is any graph, then the total degree of G equals twice the number of edges of G: the total degree of G = 2 (the number of edges of G) Corollary 1: The total degree of a graph is even. Corollary 2: In any graph there are an even number of vertices of odd degree. Application to an Acquaintance Graph: Is it possible in a group of five people for each to be friends with exactly three others?
Terminology of Graph: Paths A path between two nodes is a sequence of distinct nodes and edges connecting these nodes. Example: a b Two nodes are calledconnected if there is a path between them. Fact: For any two nodes a and b of a graph, there is an efficient way to determine whether a and b are connected or not.
An application of graphs in solving a puzzle From an initial position on the left bank of a river, a ferryman wants to transport a wolf, a goat, and a cabbage to the right bank. Ferryman s boat is only big enough to transport one object at a time, other than himself. For obvious reasons, the wolf cannot be left alone with the goat; the goat cannot be left alone with the cabbage. How should the ferryman proceed?
An application of graphs in solving a puzzle To solve the puzzle, create the following graph: Create a node for each allowable arrangement. E.g., ( fg | wc ) is an allowable arrangement since the ferryman and the goat are on the left bank, and the wolf and the cabbage are on the right bank. Create an edge between two nodes if it is possible to go from the arrangement of one node to the arrangement of the other node by a single ferry trip. E.g., there is an arc between nodes ( fgw | c ) and ( w | fgc ) because the transition from the first node to the second node can be realized by a single trip of the ferryman with the goat from the left bank to the right bank.
An application of graphs in solving a puzzle The resulting graph is: fwgc | fwg | c fwc | g fgc | w fg | wc c | fwg | fwgc wc | fg w | fgc g | fwc To transport everything from the left bank to the right bank, we need to find a path from node ( fwgc | ) to node ( | fwgc ) in the graph. There are two this kind of paths. One of them: (fwgc | ) (wc | fg) (fwc | g) (w | fgc) (fwg | c) (g | fwc) (fg | wc) ( | fwgc)
